A must move game is a game that acts as an overflow game for players from a full main game. Players in a must move game are required to switch tables from the must move to the main game when a seat opens at the main game (hence the name "must move").
In very small card rooms, there may be only one table of a given variant or limit running at a given time. If enough other players show up to start a second game, the management may decide to start a second game to accommodate them -- but they do not want to run the risk of empty seats opening up at their main game without players on a waiting list ready to fill them; most players at small casinos prefer to play at games with full tables and don't much care for playing at shorthanded tables.
In such cases, management starts a must move game, where players can play at the secondary table but are forced to move into the main game when seats open. This keeps the main game as full as possible and keeps management from having to "merge half-empty tables" if the two games simultaneously wind down.
Must move games are nearly always the same variant and limit as the main game to which they are connected, but not always. At the smallest card rooms, players at each table decide which variant to play, so if the players at the must move decide to play Omaha/8, they can, even if the main game is 3/6 limit hold 'em.
Generally, players are moved from the must move table to the main game in the order in which they arrived at the must move table (first in, first out), though different casinos may use different rules (e.g. player under the gun, or player in the cut off seat). Also, in most casinos, the player in question has the option to turn down the move and pass it on to the next player in line at the must move, if they so desire. Players on a lucky streak, or who believe the must move is a weaker table than the main game, may decide to try to stay at the must move table. Casinos generally allow this, since they don't really care who moves, as long as someone does, but if every player at a must move refuses to move, the casino will generally enforce a house rule that dictates which player must switch tables.