Standard or Normal Angle
Standard angle lenses are typically considered to be between 35mm and 100mm on a full frame sensor. Standard lenses are great for everyday shots, portraits, street photography, wedding, events and studio photography.
Prime
A Prime lens is any lens that has a fixed focal length - meaning that it does not zoom in or out at all. A 50mm f/1.8, an 85mm f/1.2 and a 100mm f/2.8 are all examples of Prime lenses. Some advantages of prime lenses over zoom lenses are that they tend to be sharper than zoom lenses at comparable apertures. They also tend to have wider maximum apertures allowing them to let more light into the camera making them great for low-light conditions.
Macro
A Macro lens is a lens that has been engineered to allow for extremely close focusing to a subject. These lenses are great for capturing fine details or small objects that would otherwise be difficult to photograph with a standard lens.
Zoom
Zoom lenses, unlike primes, are lenses that have a variable focal length - meaning that they can be zoomed in or out to get closer to or further away from a subject. A 70-200mm f/2.8, an 18-55mm f/4.5-6.3 and a 24-105mm f/4 are all examples of Zoom lenses. The biggest advantage to a zoom lens is its versatility. Unlike prime lenses, zooms can cover many focal lengths in one - where you may need multiple primes to cover the same range.
Wide Angle
A wide angle lens is typically considered anything wider than about 35mm on a full-frame sensor. On crop sensor cameras, the equivalent would be around 24mm. Wide angle lenses are great for landscapes, real estate, architecture and more.
Telephoto
Telephoto lenses are usually considered to be any lens that is longer than about 100mm. These lenses are perfect for wildlife, sports, concerts, events and can even be used for portraits.