A couple years ago I had problems with my driver side window when it would roll up really slow ('96 Civic). I went to the dealership to get a new motor in there, which I guess was getting old. Now I'm experiencing the same thing on my passenger side window. If I remember correctly it was somewhere around $150 to get fixed.
What exactly is the name of the part that is having problems? Is this something that I can fix myself?

 

If the window can still move up and down but does so very slowly, then:

1) Purchase a can of silicone lubricant
2) Roll the window all the way down
3) Spray some lubricant into the two rubber side channels where the edge of the window slides. Start from the top of the channel and work your way down to the top of the window glass. The lubricant will drip down the entire length of the side channels into the door.
4) Roll the window up and down several times.

If this procedure works, you should notice that the window now moves more quickly.