While there are more types of microphones, the three useful for podcasting are dynamic, condenser and ribbon microphones.
Condenser microphones have been used for all kinds of applications for a long time, and quality varies a lot. They require some sort of power source, like phantom power. Dynamic microphones are often more directional than condensers (which can use different pickup patterns). Dynamics are also often more robust. They are the most common microphone for live use, but are used in studios as well. Ribbon microphones almost seemed to disappear for a while but have seen increased usage the last years. They typically use a figure-eight pickup pattern, meaning that they pick up both at the front and the back.