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In the words of Euclid:

A surface is that which has length and breadth only.

(The Elements: Book $\text{I}$: Definition $5$)


An extremity of a solid is a surface.

(The Elements: Book $\text{XI}$: Definition $2$)

Plane Surface

In the words of Euclid:

A plane surface is a surface which lies evenly with the straight lines on itself.

(The Elements: Book $\text{I}$: Definition $7$)

The Plane

The plane is the term used for the general plane surface which is infinite in all directions.