The Disabled list is a fairly basic concept. When a player gets hurt, and he is going to be hurt for an extended period of time, you move the player onto the disabled list. There are two different lists, the 15 day, and the 60 day.
15-day Disabled List
This is used to allow teams to bring a player onto the 25 man roster. Any player placed on the 15-day disabled list remains on the 40 man roster. A player can be moved from the 15-day to the 60-day, but not vice-versa.
60-day Disabled List
This is used to allow teams to bring a player onto the 40 man roster. A roster spot is freed up by placing a player on the 60 day disabled list.
In both cases, a player may be placed on either list retroactively up to 10 days into the past, providing that they have not appeared in a game since. Also, players are required to be on the list for the full amount of time (either 15 or 60 days).
Players may also be sent down to the minor leagues for rehabilitation, but there are limits of 20 days for position players and 30 days for pitchers.