A frame is a part of a beehive that usually slides down into a wooden box and allows the bees to have something to build cells on. Cells are where the bees store food (see Pollen and Nectar for more information) and also where bee eggs are laid (see Brood for more details).
In some types of hives frames can be a wooden part that holds another piece called foundation. Foundation is usually a plastic piece that has a set cell size, this forces the bees to follow a uniform pattern instead of the chaos that they usually make.
This is a deep sized frame from a langstroth hive. See Boxes for details on size.