In Azeroth, players will encounter many different factions throughout the world. By completing certain quests or killing certain creatures, players are able to gain favor with some of these factions. Gaining reputation with certain factions allows players to purchase items or undertake quests that are unavailable to other players.

Reputation Options

You can click on a faction in the reputation pane to get additional details/options.
You can display one of your reputations on your main display where your experience bar currently displays. If you are not level 60 it displays two thin bars - experience and reputation.
You can mark any of your factions as "Inactive". These factions will be moved to the bottom of your reputation pane in a callout that is closed by default.

[Reputation "Experience" Bar]
How can I change my reputation?

Complete certain quests
Kill certain monsters
Factions

The Horde and Alliance are each considered a team, and the four races comprising each team are considered factions. The four starting racial factions for each team are displayed under the appropriate team's header.
The factions of the opposing team are always at the lowest reputation level of Hated, and are always locked in the "At War" state.
Players have a bonus to their starting reputation with their own race's faction.
Undead Forsaken players have lower-than-normal starting reputation with their team's Troll, Tauren, and Orc factions.
Troll, Tauren, and Orc players have lower than normal starting reputation with their team's Forsaken faction.
You cannot declare war on any of your own team's factions.
Other Information

The "At War" check box allows you to attack members of a faction if they are not hostile toward you. Check that box and you can now attack them.
You get a 10% discount at high reputation for buying items, item repair, training costs, and taxi flights.
All members of both the Horde and Alliance are reporting low stockpiles of various textile resources! Donations of cloth are now being accepted in the following locations: Darnassus, Stormwind, and Ironforge for the Alliance; Orgrimmar, Undercity, and Thunder Bluff for the Horde. Seek out an official Cloth Quartermaster for more information. You can use these quests to increase your reputation with various major factions to allow you to ride mounts of another race.
There is a spell effect when you increase your reputation to a new level.
The Reputation tab displays on mouseover exactly how many reputation points you need to attain the next reputation level with that faction.