That may vary based on your school, but our four primary ones are:
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Usually, four. The fighting stance, the front stance, the back stance, and the Kibadachi, or horse stance. Depending upon your definition of "basic", this may or may not also include a cat stance.
It will depend on the style of karate. We learn the front stance and back stance first. Kibadachi is usually taught as a part of warm ups. Naihanchi stance would be the next, then crooked cat and long front stance.
There are many different disciplines of karate. There are two many to mention. However they all come down to same same purpose, the way of the warrior. (bushido)