An independent claim stands alone. For example, if you had invented the bicycle, an independent claim might look something like this:

   1. A device for transportation consisting of two
   wheels joined by a frame on which one or more
   passengers sit.

A dependent claim depends on an another claim. Here's an example that depends on the above independent claim:

   2. The device of claim 1 where one of the wheels is in
   the front of the vehicle and one of the wheels is in the
   back of the vehicle.

Note how it starts with claim 1, and then further refines it.

Here's an example of a dependent claim depending on another dependent claim:

   3. The device of claim 2 where the front wheel is mounted
   in a turnable fork for steering the vehicle.