What is a relative integral basis? Reviewing some number theory, particularly relative discriminants for extensions of number fields, I'm running into a common problem where when I look for the definition of a term: all I can find are people's research papers online which assume you already know what it means!
So, as far as I know, if you have a finite field extension $K\supset\mathbb{Q}$, and denote $R_K$ for the ring of integers of $K$, then an integral basis $\{\beta_i\}$ of $K$ over $\mathbb{Q}$ is a basis for $K$ over $\mathbb{Q}$ in the usual linear algebra sense, with the additional properties that it's contained in $R_K$, and it is a basis for $R_K$ over $\mathbb{Z}$.  (So the word "integral" is reflecting how it corresponds to a basis over the integers.)  You can then define the (absolute) discriminant of $\{\beta_i\}$ to be $det(\sigma_i(b_j))_{i,j}^2$ where the $\sigma_i$ are the members of $Gal(K|\mathbb{Q})$.  And we get that this is independent of choice of basis, hence well defined as the discriminant of $K$ over $\mathbb{Q}$.
But then, let's suppose we have another finite field extension $L\supset K$.  One can find talk about a "relative integral basis" for $L$ over $K$, and conditions under which this may or may not exist.  But before I can read on, I need to know what a relative integral basis is.  Since none of these books I have tell me, I would assume it's the natural analog defined like this:
$\{\beta_i\}$ should be a "relative integral basis for $L$ over $K$" iff
(i.) it's a basis for $L$ over $K$ in the usual sense,
(ii.) $\forall i:\beta_i\in R_L$, & (iii.) it's a basis for $R_L$ over $\mathbb{Z}$.
But then the next paragraph in the literature says that since the "relative integral basis" may not exist, we define the relative discriminant of $L$ over $K$ as the ideal in $R_L$ generated by the discriminants of all bases having the form of the one I just thought was the relative integral basis!  How confusing.
So is the "relative integral basis" really a basis for $L$ over $K$ such that it's in $R_L$ and is a basis for $R_L$ over $R_K$?  Like "relative" as in over the "relative integers" from the other ring of integers?  Now that I've typed all this out, it seems that's probably the answer... if so I guess all I need is a "Yes" from a knowledgeable person and I can carry on.  If not, please set me straight.  I sometimes get really lost in the terminology and notation of algebraic number theory; sometimes I wish we would just rename and rewrite everything from scratch in a more intuitive way.
 A: Your "question" seems to contain lots of general grumbling about the deficiencies of the standard notation -- well, sorry, these are hard concepts, and no amount of tinkering with the terminology is going to magically make them transparent. 
A relative integral basis is, as you guessed and Qiaochu confirmed, a basis of $\mathcal{O}_L$ as an $\mathcal{O}_K$-module, which exists exactly if $\mathcal{O}_L$ is free as an $\mathcal{O}_K$-module (which isn't always the case). The other conditions you suggest in your last paragraph are automatic from this -- any $\mathcal{O}_K$-module basis of $\mathcal{O}_L$ is automatically contained in $\mathcal{O}_L$, and is a vector space basis of $L$ over $K$.
These don't always exist, as we've seen; but there are always vector space bases of $L$ over $K$ which are contained in $\mathcal{O}_L$; and the definition of relative discriminant involves an ideal generated by all choices of subsets having this strictly weaker property. 
Of course, if relative integral bases exist, then the relative discriminant ideal is exactly the principal ideal generated by the discriminant of any choice of relative integral basis. (Try to prove this; it's a good exercise.) 
