I'm afraid I have no idea who all of them are.

I'm just honest, I never really got into animes. Which one would you recommend to convince me?
It depends really, as animation is really a HUGE medium in Japan. There's science fiction, fantasy, comedy, horror, action, drama/slice of life, sports, romance, etc. What's you're favorite genre of fiction?
As for my personal opinions:
1.
Monster: (Drama, Mystery,Horror) It follows a skilled Japanese neurosurgeon named Kenzo Tenma working in Germany who, after being fed up with his boss playing hospital politics, saves the life of a more critical patient (a boy shot in the head) and as a result loses his upcomming promotion and his fiance. Later on, his boss and coworkers are killed which has the effect of making Tenma the hospital director and also making him the prime suspect. Framed for crimes he hasn't committed he becomes a fugitive in order to find the real killer, chased by a workaholic German BKA agent, a hired killer, and his crazy ex-fiance. Not available commercially in the US and quite long at 74 episodes but highly worth watching.
2.
Ghost in the Shell: Stand Alone Complex: (Science Fiction, Crime Drama) In a unique future world where the lines of man and machine are no longer as clearly defined, a covert branch of the Japanese government called Section 9 deals with cybercrimes, terrorists, and other crimes related to cyborgs and androids. The team consists of cyborgs all with their own strengths and skills. Runs 52 episodes and has been fully released. This series has been aired on Cartoon Network and might still be or come back sometime since it got good ratings.
3.
Planetes: (Science Fiction, Drama, Comedy, Romance) In the not so distant future space travel between our little part of the solar system has increased. This has the added effect of of making Earth's orbit filled with random space junk from launches which pose a very real danger to incomming and outgoing flights. The debris section dubbed "half section" consists of a team of Space EVA workers that collect, recycle, or destroy the debris. It's probabally the best and most believable hard sci-fi show that I've seen. It actually pays attention to things like physics. 26 episode series available for a very decent price.
4.
Haibane Renmei: (Drama, Mystery) Confused and without many prior memories a girl emerges from a strange cocoon into an unfamiliar walled city to which she is not allowed to leave. Her only memories of who she might be are pieces of a dream of falling from the sky. Because of this she is given the name Rakka which means "Falling" and soon short angelic wings sprout painfully from her back. She learns from the girls like herself that take her in that she is a Haibane or "charcoal feather". Rakka attempts to adjust to her new strange lifestyle, the city's strange laws, and solve the mystery of the town's purpose, and how she got there. Short and sweet at only 13 episodes. The company that put it out, Geneon, died last year though so finding DVDs could be difficult.
5.
Black Lagoon: (Action) Rokuro Okajima is an ordinary Japanese Bussinessman fed up with the crappy way he gets treated. However, he finally get a chance and is charged with delivering a disc which is where his troubles really begin. First his boat is hijacked by modern pirates, calling themselves Lagoon Company who were paid to recover the disc which their employer plans to use to blackmail his company because it contains info on their illegal dealings. Unfortunately for "Rock", as his captors have hanceforth dubbed him, his company refuses to pay his ransom and sends mercenaries to kill him and the pirates. Left with nothing but the clothes on his back and far from home Rock decides to join Lagoon Company and become a criminal. Runs a total of two 12 episode series for a total of 24. Another victim of Geneon's colapse so the DVDs would be rare. It airs on Starz Edge though if you get premium channels.