All this data is potentially out of date, and should be taken with a truckload of salt
A capital ship is any of a class of ships requiring the Capital Ship Assembly Array to construct.
A Brief History
The presence of capitals on the battlefield has long been a source of acrimony. Prior to their introduction, a "deathstar" POS was effectively invulnerable, meaning that low-moon-count systems (where a deathstar could be deployed at every moon) were regarded as ideal places for new Outposts. A group of two dreadnoughts in siege mode, however, were able to easily tank such a structure. Capitals were therefore regarded as "game-breaking", and every alliance rushed a capital fleet into production and training lest they be left out of the capital age. Players soon noticed the tremendous hauling capacity of carriers, and began using them as a gatecamp-proof way of supplying their favorite 0.0 haunts. Meanwhile, motherships and titans were regarded as nearly-unbuildable, with only one having been created, a Nyx-class mothership, at the cost of dozens of POS's, an outpost, and an alliance's entire 0.0 presence (see The Battle of EC-P8R).
With continued expansion of the ranks of capital pilots, however, the first cracks in the eggshells upon which capital pilots walked started to become visible. Carriers began to be deployed on the front lines of battleship-class battles to great effect, while fleets of ten, twenty, or thirty dreadnoughts could wipe out a hardened tower in less than thirty minutes. With CYVOK's unveiling of his Avatar-class Titan, and an extensive mothership-building program underway in both ASCN and BoB, supercapitals lost their "unbuildable" status. CYVOK's Avatar was soon matched by Shrike's, and from there the numbers of capitals and supercapitals on a typical battlefield began to explode. Shortly thereafter, a massive buff to capital ship hitpoints meant that dreadnoughts went from fifty or a hundred thousand HP to a quarter to half a million, and with the possibility of dying due to warp-in lag or while waiting for a rep to finish greatly reduced, the capital age was here to stay.
Types of Capital Ships
Capital ships fall into four lines, only one of which has more than a single ship in it for a given race.
Carriers are multi-functional vessels, serving on both the frontlines and in a logistics capacity. Supercarriers (formerly known as Motherships) , their supercapital cousins, are in most ways simply a scaled-up carrier, but with the ability to field Fighter-Bombers giving a fearsome anti-capital capability.
Titans are the largest ships in Eve, capable of one-shotting a carrier or dreadnought (hopefully an enemy one, but not always!) with a single activation of their devastating doomsday weapon, or of bridging the vast chasms of 0.0 space with their Jump Portal.
Rorquals are the ugly ducklings of the capital family, being confined to a single ship instead of one for each race. They are intended as mining support par excellence, having the unique ability to compress ore for easy transport.
For further details relating to the module changes in particular, these were announced by CCP Nozh in this DevBlog Capital Ships in Dominion
Capital ships are also unique within Eve in that they cannot use the jump gate network by means of which (nearly) all other ships must travel. Instead, they rely entirely on their jump drive to get them to their destinations. However, once in a system, they are able to warp, and to travel at subwarp speeds, like a normal ship.
Only capitals are able to fit capital-class tanking modules, the Capital Armor Repairer I and the Capital Shield Booster I. These modules, alongside the towering hitpoint buffer with which capital ships start, mean that, typically, no single enemy ship, excepting Titans and Supercarriers, can destroy a capital ship fit for tank without calling in backup.