There are several approaches.
If you have a circle that is a triangle fan and you want to give it rings, just Circle Select C the edges that make up the "spokes" of the circle, Esc out of Circle Selection, and Subdivide W > "Subdivide".
In the Operator Panel on the left you can adjust the number of subdivisions.
Creating a triangle fan
If you don't yet have a triangle fan and need to create one, select the edge loop of your circle, Extrude E, then Scale S inward. Type "0" to scale in to the absolute center, Enter to commit. Remove doubles W > "Remove Doubles".
Note that the face normals may not be pointing in the direction you want. In the GIF above this is evident from the dark shading. You can remedy this by selecting the faces of your circle and pressing W > "Flip Normals".
Starting with a triangle fan
When you create your circle there is an option for Fill Type. Set it to Triangle Fan and you will already have a circle made up of tris.
A less common approach, but good to know about.
If you have a circle that is an n-gon, with the face selected, press I and you can inset it inward. Finally, Scale S in to "0" and Remove Doubles W > "Remove Doubles".
Beveling the center vertex
If you're starting from a triangle fan, one more approach you can take is to bevel the center vertex to create the edge ring necessary to perform a Loop Cut. If you type in a near-zero value the 8 resulting center verts will be right on top of each other and you can easily patch the center hole later using Remove Doubles.