iMovie will only import codecs that it can edit. imovie can import MP4, DV, Apple Intermediate Codec, h.264, Photo JPEG, Motion JPEG, Apple Animation, and perhaps others.
MOV is a container, not a codec. It can contain any of the above codecs, but it could contain other codecs that are not editable, such as MPEG 1 or MPEG2.
Similarly, AVI is a container, not a Codec, so saying you have something in AVI does not really provide many clues as to how you might convert it.
One way to move forward is to see if your file will open in QuickTIme Player. If it will, click Window/Inspector and see what is listed next to the word "Format". These are the audio and video codecs. (Did I mention that the Audio codecs must be editable as well?) Then it might be possible to give you some specific advice.
Also, iMovie will not edit MOV files that have a closed caption track, a chapter track, or a tween track. These must be removed before importing into iMovie. If these exist, you would see them when you open the QuickTime Inspector.