Mini-DV tapes are what you see in consumer camcorders. DV tapes are much larger and, as far as I know, have never been used in consumer camcorders. I think the full sized DV tapes are almost 5 inches long and about 3.5" wide. Mini-DV tapes, on the other hand, are something 2.5" by 2". They both use the same recording format (DV) and the same 1/4" tape. With a full sized DV tape, you could record about 4 1/2 hours compared to 1 hour with a mini-DV tape.
There are other variants, including DVCAM (Sony pro-gear) and DVCPRO (Panasonic pro-gear). Sony also used to make camcorders that recorded in DV format on 8mm tapes and called it Digital8. There is also a new format called HDV. It's not all that similar to DV but camcorders that use it record to mini-DV tapes.
When you talk about video formats, you're really talking about two different things - the way the information is encoded and the media on which it is written. The actual information stored on a Digital8 tape is identical to the information on a mini-DV tape and totally different than the information written on an old 8mm or Hi-8 tape.
It gets really tricky these days because you have mini-DV camcorders, compact flash card camcorders, secure digital card camcorders, mini-DVD camcorders, and hard disk camcorders. While the mini-DV ones generally all use the DV format, the others typically use either MPEG2 or MPEG4. All of that is for standard definition camcorders; it gets more complicated when you start looking at HiDef.