The axis of a cylinder is the line that connects the center points of the circular ends.
Other curved shapes can be the ends of general cylinders, but they will be named specially. A "cylinder" almost always means a right, circular cylinder.
The surface area of a cylinder is equal to the area of the bases plus the lateral surface area
The lateral surface area is the area of the portion of the cylinder that "wraps around it". You can find this area by multiplying the circumference of the bases by the height of the cylinder.
The volume of a cylinder is the area of the base times the height of the cylinder.