You have size classifications and role classifications but be warned that the games bother more these than does canon.
size classes:
Frigate: small vessels, scouts, escorts etc.
Light Cruiser: not the smallest ships, but on the smaller side of midpoint.
Heavy Cruiser: Most iconic capital ships.
Dreadnaught: big un
Now, these are relative to the rest of the fleet and thus are relative to era. For example, the Constitution class is often given as a Heavy Cruiser however by TNG standards it would be a Frigate, maybe a Light Cruiser arguably. Allthough we don't see larger ships in the TOS era so it could have been considered a Dreadnuaght while in the games the iconic hero-ship is usually from my experience designated a Heavy cruiser.
However, I'm sorry the Galaxy class is a Dreadbnaught, not Heavy Cruiser. Also I hate it when people add a third naccelle and call it a Dreadnaught. As for the "All Good Things refit" that doesn't make the standard Galaxy a HC- as that refit isn't indicative of the standard timeline fleet.
So then you get role specific classes like:
Battlecruiser: A LC or HC outfitted and being used militarily.
Explorer: Usually, if it is smaller it is a science vessel, if it is larger, having the resources to care for and defend itself on a five year mission it is an Explorer.

Now, these explorers, tend to be armed just as well as the battlecrusiers so.... granted they are further from home and not focused just on battle so they end up being a bloated, scientifically rounded battlecruiser.
Battleship: A weapons platform of a ship, well armed and intended to in the thick of it. Where Cruisers are patrol ships used to maintain territory, the Battleship is positioned where firepower is needed to enforce authority and deal with severe threats. Size isn't inherent in the name but it either tends to be a) a Dreadnaught b) a smaller ship an admiral likes and has beefed up or c) The Defiant.
Destroyer: a fast battlercruiser, weapons heavily focused forward, set to quickly overwhelm and take out lesser, even multiple opponents. Imposing, stalks and doesn't expect someone to approach it from behind. The Defiant fits this as well. Sometimes called a Striker if it has a limited payload.
The Sovy is clearly a Battleship, however it can also perform and often does, as an explorer. I consider it more battleship than explorer but this is Starfleet so of course they are going to called it an explorer with type II being the citation.
and then you have the Akira which I just love because it can be considered a torpedo boat and through-carrier which you don't see often.