I find myself becoming confused whenever I try to think about this. In the following, $K$ is a field.
An elliptic curve $\mathcal{C}$ is defined to be a nonsingular projective cubic curve over $K$, with a $K$-rational point. Now, I am told that the specified $K$-rational point is always the identity $\mathbf{o}$, which is the point at infinity, but I don't understand this. Isn't a $K$-rational point by definition supposed to be an affine point? I thought that $K$-rational means an element of $\mathbb{A}^2(K)$. The point $\mathbf{o}$ is simply not in affine space!
Another problem I have, which might stem from the above misunderstanding, is what the Mordell-Weil subgroup $\mathcal{C}(K)$ actually is. I understand that the points on $\mathcal{C}$ form a group under the usual law. But if the point at infinity is $K$-rational, and clearly all other points are affine and on a curve over $K$, then isn't the Mordell-Weil group just going to be all the points on $\mathcal{C}$? How is it ever a proper subgroup?