Weak and Strong Collision Resistance, again Weak collision resistance (CR), or second-preimage resistance, is the property that given $x$ and $h(x)$ ($h$ a hash function) it's difficult to find $x' \neq x$ such that $h(x') = h(x)$.
Strong CR, or just collision resistance, is the property that it's difficult to find any two $x,x'$ with the same hash value.

The latter is easier than the former; the former is the "hash function analogue" of "same birthday as me" and the latter the analogue of "any two persons sharing a birthday".

Now then, what would a collision attack look like? I don't understand when it would be of relevance that it's relatively easy to find a pair $x,x'$ with the same hash value; isn't an adversary typically confronted with a given hash value that he has to "work with"? I.e., $x$ and $h(x)$ are given, if he can find a collision pair $(y,y')$, well, what does that help him?

Also, how does one actually find "any two" inputs with the same hash value? Seems to me that you must always first choose one input $x$, and then compare against that, at which point we're back to finding second preimages, right?
There are many situations where collision resistance is exactly the property that you need. For example, assume that I want to get a certificate from a CA with the URL www.google.com so that I can play a fishing attack against Google. Assume that I manage to find two certificates with the same hash; one has URL that is www.google.com and the other has something else (e.g., www.yehudalindell.com). Then, the CA will happily sign on the certificate for the latter URL, and due to the collision I will now have a valid certificate for google. Another example is code signing: if I can construct two pieces of code where one is malicious and the other is perfectly valid but both have the same hash, then I can get my valid code accepted to Apple Store or something like that, but then get people to download my malicious code.

It is really important to understand that birthday attacks on hash functions are just the generic way of finding collisions when nothing better is known. However, advanced cryptanalysis can do amazing things and find really valid collisions. For example, using advanced techniques and MD5 collisions, a rogue certificate signing certificate was obtained. The Flame virus also propagated by using an MD5 collision. So, this is not there, but reality!

Weak collision resistance, or more exactly, target resistance or 2nd-preimage resistance is important when a collision must be found with a given hash. But this is often not the case.