But when I take the SD card out of the camera and put it in the computer, there are no images saved.

It may be that you have a lower cost model which does not allow image/video storage. On these models, the Capture button simply freezes the video. Press any button to go back to live video.
A memory card may be included with the camera, even though the camera cannot save images, so that it is available if you decide to upgrade to the other model.