The different classes of ships are:
Frigate, Destroyer, Cruiser, Battle Cruiser, Battleship, Capital, Super Capital and Auxiliary
Frigate-These are the smallest player piloted ships in Eve and there is a lot of versatility built into the class. Some frigates are specialists with bonuses and slot layouts designed to fill a specific purpose, and some are generalists with enough room for different types of fitting based on individual preference. Frigates are able to operate solo, in small groups, and as part of a large fleet. Both T1 and T2 frigates follow this pattern.
Destroyer-The destroyer class is the first of the graduated progression levels and is designed around the idea of doing damage. The problem with destroyers is that they use frigate weapons in large amounts instead of using specialized weapons. In old earth history destroyers were used against submarines, fitted with guided missiles for long range engagement, and equipped with torpedoes for use against larger foe. The Eve destroyer falls short of the class's potential for T1 hulls, but does a good job with interdictors at the T2 level. There are a few niche areas for T1 destroyers to excel in and those roles are almost always as frigate killer.
The cruiser class in Eve is a group of specialist hulls designed to fill a specific role in a fleet, but with almost no ability to diversify. Each cruiser has bonuses and fitting layout with a single basic intent for that hull to perform. There are combat cruisers but they too are rather limited in options for customization. That said, if you need a cheap ship to plug a specific gap in your fleet composition the cruiser is a great choice. Cruiser can operate solo or in small groups, but they really shine in large fleets performing their bonused role.
T2 Cruiser- The specialization of the cruiser class has resulted in the T2 cruiser line becoming incredibly effective at their role. Recon, Logistics, Heavy Interdiction, and Heavy Assault are all excellent for fulfilling the intended role. The only weak point in the T2 cruiser line up is Heavy Assault which offers great offensive capability in a cruiser sized ship but in many cases a Battle Cruiser would offer just as much fire power while also providing a well rounded defense.
T3 Cruiser- The cruiser class has the distinction of being the only hull in Eve that has a T3 variant. The T3's fill a multitude of roles and offer customization unequaled by any other class. This tier of ships excels in wormholes especially.
Battle Cruisers (BC) are the work horse of New Eden and the second graduated progressiom level. They offer a combination of firepower, defense, fitting options, slot layout, and affordability that have made them the ship of choice for a wide variety of tasks. BC's can run missions, engage in PVP, and kill off incursions with very little time spent in training and with a high degree of effectiveness. T2 BC's provide command links that boost the entire fleet and also provide a very tough, if expensive, combat ship for those not in a command position.
Battleship-The Battleship (BS) used to be the dream very pod pilot aspired too, however over the years changes have been made to reduce the popularity of the BS. A capable platform, most often seen running level 4 missions or fighting off Sansha incursions, the BS still has a definite role to play. BS's are equipped with powerful weapons capable of engaging other BS size ships as well as Capitals. The biggest downfall of the BS is it's speed. Slow both on grid and getting into warp, the BS is best utilized in a fleet of similar sized ships with cruiser support. The BS fleet is capable of engaging a wide variety of targets and is still one of the backbones for PVP in nullsec, but BS fair poorly in small numbers or solo due to the likelihood of getting ambushed by smaller, faster opponents. T2 BS are very unique and require a significant training commitment so ensure your play style can benefit from the advanced BS before making the time investment.
Combat Capitals come in two flavors; Carrier and Dreadnaught.
The Dreadnaught is the big brother of the BS and uses the largest weapons in the game to rain destruction down on stations and other capitals. The problem with Dreadnaughts is they are even slower than BS's and highly susceptible to attacks from Super Capitals. 'Naughts shine in POS attacks however and are routinely used to bring down stationary space modules.
Carriers are logistics ships on steroids while still packing a fairly mean punch. The role of a Carrier is to act as a mobile repair base as well as being the first ship to carry fighters. Fighters are highly capable drones with unique usage rules and are effective against anything smaller than a Capital ship. Carriers can be customized for several different tasks to include missions, ratting, combat, and fleet defense. Carriers are to Capitals what BC's are to sub-capital ships.
Super Capitals also have only two options; Titans and Super Carriers (Moms).
Titans are the father of the Dreadnaught and BS, offering the single most destructive weapon in the game, the Doomsday device (dun dun dun!). Titans can equip the same weapons as a Dreadnaught but offer greater levels of defense, more overall utility, and the ability to do massive amounts of damage to a single ship.
Super Carriers were originally called mother ships (Moms) and are the only ship to utilize the fighter bomber (FB). The FB is effective against Capital ships and a Mom can launch a staggering number of these deadly drones. Currently there is no effective counter for a Mom except a greater number of Moms.
Auxiliaries are any ships designed to support and augment a fleet while having no true combat role. Examples of auxiliaries include industrial transports, mining barges, and salvage vessels. There are Capital class auxiliaries as well; freighters, jump freighters, and capital industrial ships. All of these vessels are designed to operate outside of normal combat however most provide some small self defense capabilities. Primarily these ships are used to support combat operations and ensure the war effort can continue.