This is the timeline of events:
2053: World War III. Limited nuclear exchange, 600 million killed directly by the fighting, major governments throughout the planet collapse. Earth begins a slower recovery from a "new dark age."
Mid-2050s: Infamous militia leader Colonel Green leads a genocidal campaign to murder any survivors of the nuclear exchange or their offspring displaying symptoms of radiation poisoning or mutations.
5 April 2063: Zefram Cochrane launches the Phoenix, a Titan V ICBM missile from a silo just outside Bozeman, Montana, converted into a small, warp-capable space capsule. The Phoenix reaches orbit and then successfully reaches Warp 1 for about thirty seconds, going out past Luna and then returning to Earth. A Vulcan survey vessel on the outskirts of the Sol system called the T'Plana-Hath detects that Humanity has invented warp drive and lands in Bozeman that evening. Zefram Cochrane himself makes first contact with Vulcan.
2163-2151: Over the next nine decades, word spreads of the confirmed existence of advanced aliens. Seeing what war and division has wrought upon the Human race -- its near-destruction -- and seeing what other species have shown themselves capable of, something fundamentally changes in Humanity. Old divisions start to be overcome. Humanity realizes it needs to get its act together. War, disease, famine, and poverty are wiped out by the 2150s.
Some time prior to 2150, the state called United Earth is founded with the intention of unifying the entire planet under a single democratic government. In 2150, the last holdouts join United Earth. The novel Articles of the Federation establishes United Earth to have been established in 2130.
Sometime during that period, the United Earth Starfleet is also founded, charged with developing advanced new warp engine technologies and with beginning the process of Humanity's exploration of the galaxy. Simultaneously, an organization called the Earth Cargo Authority is founded, operating a fleet of very small trading vessels that establish commercial ties with many nearby worlds. The operators of the ECA ships are called "Boomers;" many of them grew up on those ships and have never known planet-bound life.
2151: The United Earth Starfleet launches Enterprise NX-01, the first Earth vessel capable of reaching Warp 5 -- and therefore the first Earth ship fast enough to make interstellar travel on more than a limited -- or generations-long -- basis practical. Enterprise makes first contact with dozens of different species, and helps establish United Earth as a legitimate interstellar power. That same year, first contact is made with the Klingon Empire, and a rocky relationship is established.
2151-2155: In the course of the NX-01's career, divisions of United Earth's elite Military Assault Command Operations (MACO) agency are deployed on Enterprise and, later, its sister ship, Columbia NX-02, in recognition of the conflict situations Enterprise keeps finding itself in. This happens as the United Earth Starfleet, facing the Xindi threat from 2153 to 2154, begins to realize that its role will necessarily evolve to include planetary defense.
As a result of the efforts of Enterprise Captain Jonathan Archer to encourage cooperation between the often-warring Vulcans, Andorians, and Tellarites, it is later discovered that a mysterious race called the Romulans have been manipulating those worlds in an attempt to provoke a war -- apparently as a prelude to invasion. Archer uncovers the Romulans' role, and United Earth Minister Nathan Samuels organizes the following January the establishment of the Coalition of Planets -- a sort of interstellar NATO, a mutual-defense alliance between Earth, Vulcan, Andor, Tellar, and, or so Minister Samuels hoped, Rigel, Denobula, and Coridan.
2156-2160: Earth and its allies end up fighting the Earth-Romulan War against the Romulan Star Empire. No Human or ally ever sees a Romulan during this conflict. When it ends after the Romulans are humiliated at the Battle of Cheron, the Romulan Neutral Zone is established in the peace treaty -- an interstellar no-mand's-land a light-year wide, behind which the Romulans retreat, not to be seen for approximately 110 years.
2161: As a result of the war, Earth and its allies recognize the value of solidarity and unity. The Federation Charter is signed by Humans, Vulcans, Andorians, and Tellarites, founding the United Federation of Planets -- an interstellar federal republic uniting these formerly warring races under a single government, dedicated to universal liberty, liberal democracy, sentient rights, the expansion of scientific knowledge, trade, and exploration. The novels establish that the recently-independent Alpha Centauri system, formerly a United Earth colony, is also one of the signatories. The ENT novel Last Full Measure establishes the Charter -- also called the Federation Constitution, and formally known as the Articles of the Federation -- to have been signed on 12 August 2161, thereafter celebrated every year as Federation Day.
With the establishment of the United Federation of Planets, the Federation Starfleet is also chartered. The Federation Starfleet grows out of the United Earth Starfleet, Andorian Imperial Guard, and the Vulcan and Tellarite space forces. The Federation Starfleet is chartered as a military organization serving the dual primary purposes of exploration and defense, while also serving scientific, diplomatic, humanitarian, and peace-keeping functions.
It is important to bear in mind that the Federation Starfleet, despite sharing a name with the United Earth Starfleet, is no more the same organization as the UE Starfleet than the United States Navy is the same organization as the Massachusetts State Navy that existed during the American Revolution.
2161-2233: The Federation continues to grow and expand, and Starfleet is at the forefront of the Federation's growth.
c. 2220: Along the line, tensions begin to seriously heat up with the Klingon Empire, leading to a 70-year-old cold war that nearly becomes a shooting war in the 2260s.
2233: The USS Kelvin is under the command of badass Captain Robert Robau and his first officer, Lt. George Kirk. Kirk's son, James Tiberius, is born in Iowa. In the alternate timeline Nero creates (seen in the newest movie), the Kelvin is destroyed, George is killed in action, and James is born in space, fleeing Nero's ship, the Narada.
From there, the timeline splits in two. The prime timeline is the one that TOS, the first 10 movies, TNG, DS9, and VOY were all set in. The new alternate timeline is the one the writers created for the newest movie, and presumably future movies will be set in it.
Alternate Timeline:
2233-2255: Starfleet is profoundly changed by the destruction of the Kelvin, adopting the Kelvin mission patch as the emblem of the entire Starfleet. A planet-based starship construction facility called the Riverside Shipyards is established outside Riverside, Iowa, the home town of George Kirk, in tribute to his sacrifice. Local Riverside bars even feature Kelvin-shaped salt shakers in celebration of George Kirk's heroism. Tensions are also higher with the Romulans as a result of the Narada incident. Romulans are also discovered to be the same species as Vulcans much earlier than in the prime timeline.
2255: Capt. Pike recruits the young delinquent James T. Kirk into Starfleet, recognizing in him command potential.
2255-2258: James Kirk is a student in the Federation Starfleet Academy, quickly rising to the rank of Lieutenant while still serving as a cadet.
2258: Launch of the USS Enterprise NCC-1701 (13 years later than in the prime timeline) under the command of Capt. Christopher Pike. The Narada returns upon detecting the ship piloted by Ambassador Spock from the year 2387. Old Spock is captured, then exiled to the planet Delta Vega in the Vulcan system. The Narada destroys the planet Vulcan and a fleet of Federation starships, leading to six billion deaths. Nero and the Narada are later destroyed during an attempt to destroy Earth by the actions of Cadet Kirk and Lieutenant Commander Spock and the mostly-cadet-crewed Enterprise. Kirk later promoted to captain and given command of the Enterprise, which begins a five-year mission of exploration.
Don't read any further if you don't want spoilers for TOS, the first ten movies, TNG, DS9, or VOY.
In the prime timeline
2230s-2260s: The Federation continues to grow and expand, and the cold war with the Klingons continues.
2245: The USS Enterprise NCC-1701 is launched under the command of Captain Robert April of Earth. The Enterprise completes a five-year mission of exploration before Captain Christopher Pike of Earth takes command, completing at least three five-year missions. Spock of Vulcan serves 11 years under Capt. Pike. The Enterprise's mission patch is based on one held earlier by the Kelvin, implying that the Kelvin was an extremely influential ship of the early 2200s.
2265-2270: This is TOS. Command of the Enterprise is handed over to Capt. James T. Kirk, with Lt. Commander Spock serving as his first officer. The Enterprise begins its most famous five-year-mission of exploration, making first contact with hundreds of species. The Romulans come out of their self-imposed isolation and are discovered to be the same species as Vulcans. At the end of TOS, Kirk's Enterprise has become so accomplished that its mission patch is adopted as the emblem of the entire Federation Starfleet.
2273: Star Trek: The Motion Picture. The Enterprise has completed a major refit, and Kirk is now an admiral serving as Chief of Starfleet Operations. Kirk and his crew re-unite to stop the alien entity known as V'Ger from destroying Earth.
2285: Star Trek II: The Wrath of Khan, Star Trek III: The Search for Spock, Star Trek IV: The Voyage Home, Star Trek V: The Final Frontier. Kirk is growing older, discovers he has a son, David. The madman known as Khan Noonien Sign returns from exile, intent on revenge against Kirk. Death and resurrection of Spock on the Genesis Planet. Death of David Marcus. Kirk and company save Earth from the Whale Probe by traveling to 1986 and retrieving a pair of humpback whales capable of communicating with the probe and getting it to leave. Destruction of the original USS Enterprise, launch of the Enterprise-A.
2293: Star Trek VI: The Undiscovered Country, Star Trek Generations prologue. The Klingon Empire, facing economic collapse, sues for peace. Kirk thwarts a conspiracy within Starfleet from assassinating the Federation President at peace talks to provoke a war. The Khitomer Accords are signed, the end of 70 years of hostilities. Decommissioning of the Enterprise-A. Kirk is believed killed later that year during the launch ceremony of the Excelisor-class Enterprise-B.
2293-2364: The "Lost Era" of Star Trek. Little is established. We know that the Enterprise-C saved a Klingon colony from a Romulan sneak attack at some point, saving what had been a fracturing peace accord with the Klingons and solidifying their status as a Federation ally by the time of TNG. Border wars with the Cardassian Union are fought. In 2311, the Romulans enter a period of self-imposed isolation again following the infamous Tomed Incident. The Cardassians and Klingons fight an 18-year cold war called the Bretreka Nebula Incident.
2364-2370: TNG. Launch of the Enterprise-D under Capt. Jean-Luc Picard. First contact with the Ferengi and Borg. Borg incursion in December of 2366, assimilation of Picard into Locutus of Borg, Battle of Worf 359 in January 2367. Klingon Civil War in 2367-2368, rise of Gowron as Klingon Chancellor.
2369: Withdrawal of the Cardassian Union from the occupied world of Bajor. Bajor invites the Federation to administer the abandoned Cardassian space station Terok Nor as Federation Starbase Deep Space 9. Discovery of the Bajoran Wormhole linking the Alpha Quadrant (local space) with the distant Gamma Quadrant. Bajor becomes an important interstellar hub as a result, even as it recovers from the Occupation.
2369-2375: Star Trek: Deep Space Nine. First contact with the Dominion in the Gamma Quadrant. Rise of the Maquis. Klingon-Cardassian War, sundering of the Khitomer Accords, Klingon-Federation War. Changeling infiltration paranoia. Annexing of the Cardassian Union by the Dominion, restoration of the Klingon-Federation alliance. The Dominion War breaks out in December 2373. Romulan Star Empire eventually joins on the Federation side, Breen Confederacy eventually joins on the Dominion side. Breen attack on Earth in 2375. Defeat of the Dominion in December 2375, occupation of the Cardassian Union.
2371: Star Trek Generations. Destruction of the Enterprise-D, final death of James T. Kirk. Launch of USS Voyager; Voyager displaced to the Delta Quadrant of the galaxy, begins long journey home.
2371-2377: Star Trek: Voyager. USS Voyager manages to come home. Fights Borg along the way.
2372: Launch of the USS Enterprise NCC-1701-E under the command of Jean-Luc Picard.
2373: Star Trek: First Contact. Second Borg attempt to assimilate the Federation. When cube destroyed in Earth orbit, a Borg scout sphere attempts to travel back in time to 4 April 2063 to prevent Earth-Vulcan First Contact. Enterprise-E crew follow them and stop them. Discovery of the Borg Queen.
2375: Star Trek: Insurrection.
2377: USS Voyager returns home after delivering a crippling blow to the Borg Collective's network of transwarp tunnels which it used to travel nearly-instantly across the galaxy.
2379: Star Trek: Nemesis. Shinzon of Remus assassinates the Romulan Senate and sets himself up as Praetor of the Romulan Star Empire with the support of part of the Imperial Fleet. Shinzon is later killed. Death of Data.
2387: Nero and Spock travel from this year back to 2233 and 2358, respectively, in the newest Star Trek film. Destruction of Romulus and Remus, the Romulan and Reman homeworlds, by the Hobus star.
