I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some projective space?

If $X\subset \mathbb P^n$ is a subvariety of dimension $m$ embedded by a linear system $V \subset H^0(X,\mathcal O_X(D))$ then the degree of $X$ is equal to $D^m$. 


See the section about intersection of chapter I of hartshorne. Intuitively degree is the number of intersection of the dimension r subvariety and the GENERIC dimension nr linear subspace if they are both in P^n. A rigorous definition was given in that section by the leading coefficient of the Hilbert polynomial. 


I disagree with "this works only for curves". Say we've already defined degree for schemes up to dimension $n$. Then use your rule to define it for schemes of dimension $n+1$. Dr. "This works only for curves" is unwilling to go beyond the $n=0$ case, for some reason. 

