Retainers in Final Fantasy XIV are basically storage and merchant NPC characters that are hired by the players to aid them in their adventures. By default each player character can hire two retainers.
A retainer comes with an extra 80 item slots for storage (note that gil and crystals are separate from this amount), as well as 8 bazaaring slots. If you give your retainer gil you can order them to use that gil to buy items in the Market Wards, and if you set up items in their bazaar they can sell them - likewise in the Wards. To access a retainer you can summon them to your side at either a summoning bell (scattered around the cities) or in a Ward.
Do note, however, that you must go physically point the location of their shop in the Ward if you want them to bazaar or buy items, and you cannot call on a bazaaring retainer from a summoning bell. Thus, if you are planning on treating your retainer as extra accessible storage only but still want to have a permanent bazaar, you must buy access to an extra retainer.
Aside from the storage space, the advantage of having a retainer bazaaring your wares is that a retainer will stay online even if your character itself is logged out.
The Hiring Process
Each of the three starting cities of Limsa Lominsa, Gridania and Ul'dah has an NPC in the local Adventurers' Guild who you can hire a retainer from. These NPCs are Ursulie of Limsa Lominsa, Gyles of Gridania and Thaisie of Ul'dah. When you talk to one of these people they will give you a list of pre-made retainers to choose from for your employ. Each city has a different list of five people in each combination of race/tribe/gender - although naturally, the combinations such as male Miqo'te of either tribe or female Highlander Hyur that are not available in character creation are not available as retainers either.
While each retainer has a default name of their own, after picking the retainer you want to hire you must give them a nickname they will thereafter be called with. No two retainers on the same server can have the same nickname.