Let me try to give a more down to earth answer:

First, it's important to understand there are a lot of algebras whose derived categories are equivalent in surprising ways. Morita equivalences (equivalences between the abelian categories of modules) are kind of boring, especially for finite dimensional algebras; essentially the only thing you can do is change the dimensions of objects.

The way you see this is that if A-mod and B-mod are equivalent, then the image of A as a module over itself is a projective generator of B-mod, and for a finite-dimensional algebra, essentially the only thing you can do is take several copies of the indecomposible projectives of B.

On the other hand, if you take the derived category of dg-modules over A (the dg part of this is not a huge deal; it's just that they're very close to, but a bit better behaved than actual derived/triangulated categories which I consider something of a historical mistake, which should be replaced with dg/A-infinity versions), this is equivalent to the category of dg-modules over the endomorphism algebra (this is in the dg sense, so it's a dg-algebra whose cohomology is the Ext algebra) of *any* generating object. There are a *lot* more generating objects than projective generators, so there are a lot of derived equivalences.

In particular, you can take your favorite finite dimensional algebra A, and the most obvious not-very-projective generating object: the sum of all the simples. Call this L.
As I mentioned, there's an equivalence $A-dg-mod = \mathrm{Ext}(L,L)-dg-mod$, just given by taking $\mathrm{Ext}(L,-)$.

Now, in general, $\mathrm{Ext}(L,L)$ is an absolutely horrible object (ask Mikael Vejdemo-Johansson about doing this for group algebras over finite fields some time), but sometimes it turns out to be nice. For example, if you start with A being the exterior algebra, you'll get a polynomial ring on the dual vector space. Another (closely related) example is that the cohomology of a reductive group (over C) is Koszul dual to the cohomology of its classifying space (here you see a hint of this delooping mentioned in Scott's answer).

One thing that could help you make sure that $\mathrm{Ext}(L,L)$ is nice is if your algebra is graded. Then $\mathrm{Ext}(L,L)$ inherits an "internal" grading in addition to its homological one. If these coincide, then $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting A-infinity operations, they would break the grading), so you're dealing with a derived equivalence between actual algebras, though you have to be a bit careful about the dg-issues. You've found that the derived category of usual modules over A is equivalent to dg-modules over B (with its unique grading) and vice versa. You can fix this by taking graded modules on both sides.

As for more examples...well, some collaborators and I found some cool examples
coming from the combinatorics of hyperplane arrangements.