While not exhaustive, these are the terms for lens types I've run into:
Prime vs. Zoom
A prime lens is a lens with a fixed focal length. A zoom lens is a lens with a variable focal length. Simple as that.
Wide, Normal, Telephoto
These designations are about the focal length that the lens has: short, medium, and long, respectively. In full-frame equivalency, wide = 24-35, normal is 50mm, and telephoto is >85mm.
Wide angle lenses will generally give a wider field of view and are often suitable for landscape photography. Normal lenses give pretty much the same magnification as your eyes do, unaided, and feel "neutral" as lenses. Telephoto lenses have "reach" and magnify objects, and are often suitable for wildlife and sports.
These terms indicate the maximum aperture of the lens--how wide the aperture can be opened up. The bigger the opening, the faster the shutter speed you can use with the lens, hence the term. Generally, f/2.8 and wider (smaller f-number) lenses are considered fast; f/4 is medium; and f/5.6 and smaller lenses are slow.
Typically means they're past the "typical" ranges. I.e., a "superzoom" lens has a very wide zoom range (typically 10x or more), a supertelephoto is typically a lens with a focal length of 400mm or longer. An ultrawide lens is one that goes wider than a typical wide angle lens.
A walkaround lens is a lens that's good for casual shooting while walking around. It's most typically going to be a wide-to-slightly telephoto zoom, such as an 18-55 kit lens for crop. But can also be a fast wide/normal prime or a superzoom, depending on the shooter's preference.
A portrait lens is (obviously) a lens for taking portraits. It is typically considered to be a fast, slightly telephoto prime, but can also encompass f/2.8 telephoto zoom lenses, or a fast 35mm lens. It all depends on the shooter's preferences.
The more exotic lenses also have specific terms to differentiate them.
Macro: Lenses specifically designed for close-up photography, they typically use a floating element or group to optimize the minimum focus distance. Very sharp lenses for the most part.
Fisheye: Lenses that do not map straight lines as straight (rectilinear), but as curves to cover a much wider field of view. Tons of distortion. Comes in two flavors: diagonal (which cover the sensor corner-to-corner) and circular (which puts the entire image circle into the frame, so yields a circular image).
Tilt-Shift: Lenses that can be shifted from the center of the camera mount to the top/bottom/left/side and that can also tilt relatively to the front face of the camera to approximate view camera movements. Used for perspective correction, viewpoint shifting, toy/model effect. Always manual focus lenses.
Mirror: Mirror lenses use a mirror instead of glass elements to achieve large magnification, typically at low cost. Very slow, often fixed aperture, and (if inexpensive) can be very soft.
Manual: Manual lenses do not communicate electronically with the camera body. Sometimes they're vintage lenses, but they can also be new lenses, such as those from Samyang. As a result, they do not autofocus, aperture must be set with a ring on the lens, there is no lens EXIF information, and stop-down metering must be used.