It depends what kind of prism:
Rectangular prism-rectangle
Could be circle too.
It depends on what kind of prism it is. If it is a rectangular prism, it's base is a rectangle. If it's a triangular prism, it's base will be a triangle. P.S.-If you have any other questions about prisms or geometry in general, feel free to ask me!
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A triangular prism can have a square or triangular base.
Triangular
You can call it a "triangular prism" rather than "triangular based prism" ... saying a prism is "triangular" imply that the base is a triangle. There's no other name that I know of. A prism is just a 3d shape, so all you have to say is <type of shape> + prism to get the point across.
Every prism has a base. For a triangular prism the base is a triangle.
It depends on the shape of the base. A pyramid or a prism can have any polygonal shape as base and the formula for its area will depend on the shape. With other solid figures, the base may simply be the face tat the object happens to be "sitting" on: roll it and another face becomes the base.