What's the difference between vanilla, french vanilla, and vanilla bean?

Vanilla bean contains visible flecks of vanilla bean in it.

“The term “French vanilla” technically describes a particular type of custard base for vanilla ice cream. This custard base is associated with a strong vanilla flavor and odor, a rich golden color, and small flecks of vanilla beans.” Wisegeek

French vanilla in ice cream also has eggs in it as opposed to regular vanilla ice cream which doesn’t. Vanilla, when it is called for in a recipe for cookies is usually “vanilla extract” which is a combination of oil from vanilla beans and alcohol. There is also “vanilla extract” that is of a lower quality that does not have any vanilla in it at all, it is usually made with synthetic flavoring agents. Here’s a little bit more info on vanilla