Gigabit routers can move data at 1 gigabit per second, this refers to the speed of the wired port usually at the back of the router. To use Gigabit on your router, you will need to use CAT6 cable. CAT5 and CAT5e will not enable data at the same speed.
With the need to move more and more data around your network, wireless and 10/100 routers may not be enough. Gigabit routers can usually move data faster through the router, and really should be a high priority for wired computers at home sharing data.
Check the list on the right for models and makes that feature Gigabit speeds.