What is a wide block cipher and why does it avoid birthday bound problems?
I don't think the term "wide block cipher" has a hard definition beyond 


  has a larger block size than the current standard algorithm(s)


which right now in most cases would equal to 


  has a block size larger than 128 bit


because AES is our current reference standard.



Now the thing with block ciphers is that they are are pseudo-random permutations and as such you can distinguish them from pseudo-random functions because you will see a collision with the latter at the birthday where you won't find that with PRPs. This can be important in scenarios where you need to generate a lot of random numbers mostly and even then I don't think the different characteristic would be exploitable in most cases.

To see how PRPs can be distinguished, assume as an example that you run CTR-mode with a PRF and a PRP, both with $n$-bit output length. You will now find that a collision in the CTR keystream occurs with the PRF at about $2^{n/2}$ invocations whereas this won't happen until after you wrap-around with the PRP.

Another application of wide block ciphers is in full-disk encryption where you can encrypt larger blocks implying that single bit-flips will destroy larger regions of plaintext thus being (hopefully) more notice-able.

The last somewhat popular application of wide block ciphers is as part of hashing functions. Imagine your average block cipher $E:\{0,1\}^k\times \{0,1\}^n\to\{0,1\}^n$, note that you can put more into the function than you take out of it. Now you expect collisions after $2^{n/2}$ invokations, ie you have $n/2$-bit collision resistance which makes non-wide block ciphers (such as AES) unsuitable for usage as a core compression function in hash functions (because $2^{64}$ operations is feasible and a similar amount of work was done for the SHA-1 collision).