The most common type is a standard curtain rod. The rod is mounted on simple brackets. The lightweight rod slips into the pocket of a curtain and is invisible once it's hung in place. These rods are both inexpensive and versatile.
These are also a nice option when you want hang two sets of drapes on one window. A double rod gives your window depth and presence and the rods have a small profile so they don't interfere with each other.
If you don't want to mount hardware on your window casings, try a tension rod.
These fit neatly on the inside of a window casing and can be moved or changed without damage to the casings.
If you have a steel door -- try a magnetic rod.
If you have a large window or door that you want to be able to open and close draperies you might want to opt for a traverse rod. These rods move on a track when you pull the cord on the side of the window.
Finally, for a dramatic look, choose decorative rods. These rods decorate the windows as much as the draperies themselves. They can be paired with finials, tie backs and decorative brackets that make the window stand out.
I hope that narrows down the choices for you -- think of it as curtain rod basics 101. I'm Mrs. Fixit, and it's just that simple.