Prime Lenses: Prime lenses have one fixed focal length that cannot be changed. They are lighter and cost less then zoom lenses, and they offer better light gathering ability. Any type of lens (macro, fisheye, tilt shift, etc) can be a Prime Lens.
Zoom Lenses: Zoom Lenses allow the photographer to vary the focal length. They make it very easy to change compositions and vary perspectives. Any type of lens (macro, fisheye, tilt shift, etc) can be a Zoom Lens.
Macro Lens: Macro Lenses are used for extreme close-up photography. Marco Lenses have a very limited depth of field when shooting objects at close proximity, so getting small things in focus can be a challenge.
Macro photography looks like this:
Tilt Shift: Tilt shift lenses are used to control perspective, in architectural photography they are used to take photo of buildings that are not warped by perspective.
The first photo in this set was taken with a tilt shift lens:
Fisheye Lens: A fisheye lens is an "ultra wide angle" lens that produces an extreme visual distortion. There are two types of fisheye, circular fisheye (which produces a circular photo) and full frame fisheye which is rectangular like regular photos. This kind of fisheye requires a larger lens.
Fisheye photos look like this.
Telephoto Lens: A telephoto lens has a physical length shorter then the lenses focal length. They are very useful for enlarging distant objects.
Portrait Lens: A portrait lens is a lens that is specifically created to take portraits. The lenses are used to shoot portraits and are in the 85mm-135mm range.