Live-action RPGs (also known as "LARPs" or "Interactive Literature") are a form of RPG which usually involve large groups of people and take place over an extended area (i.e. larger than a room). As with all role-playing, the player takes on the role of a fictional character (or sometimes a dramatised version of a real person). The term "live-action" does not necessarily mean that all of your character's actions are physically acted out. However, in general you as a player will at least walk around the game area, representing where your character is. You can then talk to other players, speaking as your character.
LARP's are distinct from table-top RPGs mainly in that there is not necessarily a referee/game-master present at any point. Thus, the rules need to allow the players of two characters to resolve things on their own -- though exceptional cases may be handled by calling a referee over. Similarly, there are rarely NPCs per se -- both protagonists and antagonists are portrayed by live people. LARPs also tend to be more effort on costumes, props, and other physical elements that enhance the mood.
LARPs also require a higher degree of organization than a tabletop game among a small group of friends. A space for the event needs to be rented or otherwise reserved. The place and time need to be organized. There also needs to be activities which can involve a large number of people wandering around the play area. Many LARPs are organized as events at gaming conventions or even LARP-specific conventions. People buy tickets which then help pay for the costs of the event.
Different LARP designs use a variety of methods for how to resolve combat or other in-character actions which cannot be safely and easily handled. There are two common categories of solutions:
Some events are based around carefully tailored situation. For example, you sign up for the event, and are then given a packet by the organizers which describes your character and what she knows. For example, you might be given the character of a young Danish prince whose father recently died. Other players are given packets describing other characters such as Horatio, Rosencrantz, Guildenstern, etc. The players can then wander around the play area representing the Danish castle, interacting and discovering clues prepared by the organizers. The outcome of the story could change wildly, however, depending on what the participants choose to do. A person dressed as the ghost of the father might walk at a specific time as designated by the organizers, but who he meets and what their conversation is will be determined by the players. These events most commonly use "theater style" resolution.
Other events are less structured in the stories and activities going on. Players can create their own characters as they wish according to the LARPs rules and guidelines. An event might be organized around two rival clans which meet at a medieval market. Players may simply wander around the market and look for good deals, or they could fight their rivals to win respect in their clan, or they could try to negotiate with clan leaders to settle the dispute. This is the more common for "live combat" resolution.
LARPs also differ in the degree of continuity. Some of the structured theater style events are single events. The next organized event will have a different setting and characters. Other LARP events are ongoing. The next event will pick up where the last one left off, and players will play mostly the same characters.
I would add in a special note on live-action play known as "freeform" and "cathartic" games in Australian gaming conventions, such as Necronomicon. In a freeform, players -- usually 12-25 -- move around a room and interact with each other in character. There are dozens of plots and subplots, each involving a handful of characters, although there may be one overarching plot line which acts as the justification for assembling all the characters in one place, such as a funeral, a party, a journey, etc. There are multiple game-masters who wander around the action, but they cannot possibly overhear all the conversations that are going on. They may have "theater style" rules for resolving the success of actions, or they may be systemless -- relying on player agreement or calling on a GM to resolve the few tricky points.
A cathartic game or module is one that was designed to produce a strong emotional reaction in the players. They are are typically systemless scenarios designed for convention play. The scenario might involve emotionally confronting subject matter, immersion in character, and avoiding anything that breaks the players concentration (such as out-of-character talk). The goal of a cathartic game is for the players to be affected by the experience (and sometimes disturbed). Occasionally games are described as cathartic when they rely on sensationalism, taboo breaking, or shock value to create a reaction.
For more information, here are some popular live-action role-playing organizations: