X-Men are part of animated shows from 1966 but there are only three major X-men animated shows:
X-Men (1992–1997) ( aka X-Men: The Animated Series): This one was the first successful launch of X-Men in an animated medium and ran for 5 seasons and 76 episodes. The show also had crossover episodes with Spider-Man (1994 TV series) and was quite successful commercially and also critically acclaimed.
X-Men: Evolution (2000–2003): X-Men Evolution was pretty well received too, but they played with the age of characters a lot, and made most of them teenagers. It ran for 4 seasons with 52 episodes.
Wolverine and the X-Men (2009) This one ran for only one season with 26 episodes and ended with a big cliffhanger which was never resolved.
All three shows work individually and have no impact on each other, so you can watch any one of the three or even all three.
There was one failed attempt at an animated show called X-Men: Pryde of the X-Men but it never made it, and only a pilot episode was created, so you can safely avoid it. The Marvel Super Heroes also has X-men in them but they never call them by that name, and their main leads are non X-men members.
For elaboration on other X-Men animated stuff and cameos refer this list.