A prime is a lens with one single focal length, eg. 60mm. A prime usually also has a fast maximum aperture.
A zoom can have, for instance a 70-200mm focal length, which means that it is the equivalent of a 70mm lens, a 71 mm lens, a 72 mm lens, and so on. Zooms usually dont have as large apertures as primes. The image quality is also usually not as good as a prime. Zooms with a large maximum aperture are quite expensive.
A macro is a prime, since it has one single focal length, and usually have quite large maximum apertures. A real macro lens can shoot 1:1, life-size. That means that the subject is as large on the cameras sensor as it is in real life, which is very useful. A zoom lens labeled "macro" usually has 1:3 or (at best) 1:2 reproduction scale, which is not as good as a real macro lens.
There are telephoto primes and zooms. Telephoto is anything longer than 105 mm (roughly). Exteme telephoto is about 400-600mm.
The best close-up images will obviously come from the dedicated macro lens, to answer your second question.
Hope I was able to answer your questions, if there´s anything more you want to know, post another reply, and I´ll see what I can do
