There is as many tattoo styles as there is artists, but usually its categorized by the area ( Japanese , Chinese style ) or by the item that is tattooed ( Butterfly , Zombie , Celebrities ) or by the technique that is used to make the artwork ( Realistic , Old School , New School )
Most famous is New School, Old School and realistic.
Old School: Was the first tattoos for common people, characteristic by its heavy colors and big black outlines. At this time the selection of colors was not as great as today and that limited the artists quite a lot. Some of this tattoos reminds of the once you can find on sailors on the 18-19 century. This style was the norm in the beginning of the 1900 to the middle of the century and today we see a big revival of this style and many artists start out making this types of tattoo then advance to other areas when they mature as a tattoo artist.
New School: Is using the same heavy outlines with black as Old school but the motives are more funny, outgoing and crazy and cannot be mixed up with traditional motives.
Realistic: This style is the most advance one, as you can hear by its name its point is to look as realistic as possible and this is achieved by great skills of drawing and making shadows.
This is a style that requires countless of hours of drawing practices with paper and pen.