(1) Two circles are externally tangent if they share a single point, and no point of one circle is on the disk created by the other circle. The circles are internally tangent if they share a point and all the points of one circle lie on the disk created by the other circle.
(2) Two circles can share a common tangent line. If the line intersects the segment joining the centers of the circle it is said to be a common internal tangent, otherwise it is a common external tangent.
See linkfor pictures.
Note that there are many cases for common tangent lines:
If the circles are exterior to one another there are 4 common tangents -- 2 internal and 2 external.
If the circles are externally tangent then there are 3 common tangents -- 2 external and 1 internal.
If the circles intersect in 2 points there are 2 common external tangents.
If the circles are internally tangent there is 1 common external tangent.
If one circle is in the interior of the other, there are no common tangents.