Crushed ice is ice broken down into small pieces, possibly along with some ice "snow". The size is roughly the size of marbles, with obviously greater variety of size and shape.
A lot of refrigerators will crush ice for you in the ice dispenser, or you can buy a manual crusher.
Crushed ice is used in drink recipes when more dilution is necessary than regular ice cubes. Because crushed ice has more surface area than ice cubes, it will melt faster, chill the drink faster, and provide more water through dilution.
You might wonder why you'd want more dilution. Some recipes make a very strong, concentrated drink. Shaking or stirring with crushed ice dilutes these drinks to achieve a proper balance.
An example is Don the Beachcomber's original Zombie recipes from the 1930's and 1950's. These drinks are extremely strong and thick before you shake them with ice. If you use ice cubes, you don't get enough dilution unless you shake for 3-4 minutes straight. Crushed ice speeds up this process and provides a nicely balanced Zombie.
Another drink that benefits from crushed ice is a Pat O'Brien's Hurricane. If you make it at home with their mix, use crushed ice (they do at Pat O'Brien's). If you make it with ice cubes, especially the big ones from a refrigerator, the drink is tooth-crackingly sweet at first and takes some time to mellow out. A Pat O'Brien's Hurricane needs dilution, so use crushed ice.
If you're using ice out of an ice machine, you're a lot closer to crushed ice than big refrigerator ice cubes. You can usually get away with using ice from an ice machine as crushed ice--you may need to shake or stir a little longer.