elliptic curve group law Let $C$ be an elliptic curve over a field $k \supset \mathbb{Q}$. Then given $P$ and $Q$, we can draw the line between $P$ and $Q$ (call this line $L$) and then "find the third intersection point", and call this point $P \ast Q$.
If neither $P$ nor $Q$ lie on the tangent line to $C$, then each $P$ and $Q$ have intersection multiplicity $1$, so by Bezout's theorem, we can find a third point, $Q$, not equal to $P$ or $Q$.
What do we take to be our third point if either $P$ or $Q$ lie on the tangent to $C$? (e.g. if $L$ is the tangent to $C$ at $P$, then $P$ has intersection multiplicity at least two and we don't get a third distinct point).
edit: $\ast$ here is not the group law, it's just the third intersection point operation. 

If the line is tangent to $C$ at $P$ and also passes through $Q$, then think of the line as passing through $P$ twice and through $Q$ once, so $P*Q=P$. That is, $P$ is the third point on the line. Similarly, $P*P=Q$ in this situation.
If the line only intersects $P$, then $P$ is a triple point and $P*P=P$.
It is probably worth noting that there is a group law on $C$ which is similar to but not quite as you have described it above. That is to say, there is a better addition rule on $C$ as follows: If $L\cap C=\{P,Q,R\}$, we define $P + Q=-R$. For reasons I won't get into here, this addition rule has much better properties.
