In Dungeons and Dragons and related systems, some actions depend on chance, with success determined by the roll of a D20. In cases where there is no downside to attempting the action multiple times (e.g., you are not in the middle of a fight), it is sometimes assumed that if the player took enough time to do 20 separate skill checks, they would roll a 20 and get the best possible result. While this is mathematically suspect, in essense, taking 20 is simply the statement that you can do any allowed, free action when the pressure is off.
Dungeons and Dragons Fourth and Fifth Editions do not allow for taking 20, although it might be houseruled in.
Taking 20 should not be confused with taking 10, which is a completely different type of D&D dicing trickery.