How do I apply differential cryptanalysis to a block cipher? I have read a lot of summaries of block ciphers particularly with regards to the NIST competitions stating that reduced-round block ciphers are – for example – vulnerable to differential cryptanalysis.

I have a general idea that the application of differential cryptanalysis is to look at the difference between inputs; Wikipedia makes that fairly clear. However, I could take any two inputs for any given block cipher and I am pretty certain I'd be staring at random differences. 


I am aware this is the idea of a well written block cipher; however, assuming a broken or vulnerable cipher (feel free to provide simple examples) how do I go about choosing differences to try? 
Are there any clues in algorithm design that would inform a decision on which values to choose? 
How does being vulnerable to differential cryptanalysis impact a cipher in the wild? 
If all I have are differences between known plain-texts and known keys as my analysis and a captured ciphertext as my data to exploit, what can I actually deduce?
Basically it's analysis of a cryptographic cypher by the means of finding a relationship between the difference in the input data and the output data.
Ideally, the slightest difference in input data (cleartext), even a single bit, should produce a completely different cypthertext. However, if the cypher is not well-designed, a correlation between the two resulting cyphertext might be observed. This correlation in turn might be exploited to find out the key.
This, obviously requires a chosen-cleartext-attack; which means the attacker should have access to the encryption mechanism, and thus use it to encrypt any number of cleartexts he or she choses. Therefore, this kind of attack is generally applied against temper-proof hardware which one could use but not get hands on the key itself.

As per AES, see this excerpt from the Wikipedia article on differential cryptanalysis:


  For example, if a differential of 1 => 1 (implying a difference in the
  LSB of the input leads to a output difference in the LSB) occurs with
  probability of 4/256 (possible with the non-linear function in the AES
  cipher for instance) then for only 4 values (or 2 pairs) of inputs is
  that differential possible. Suppose we have a non-linear function
  where the key is XOR'ed before evaluation and the values that allow
  the differential are {2,3} and {4,5}. If the attacker sends in the
  values of {6, 7} and observes the correct output difference it means
  the key is either 6 xor K = 2 or 6 xor K = 4, meaning the key is
  either K = {2,4}.
  In essence, for an n-bit non-linear function one would ideally seek as
  close to 2-(n-1) as possible to achieve differential uniformity. When
  this happens, the differential attack requires as much work to
  determine the key as simply brute forcing the key.


Here are a few links for warming up:


A simplified AES algorithm and its linear and differential cryptanalyses
Impossible differential cryptanalysis of Mini-AES
Low Complexity Differential Cryptanalysis and Fault Analysis of AES


And here are a couple of better articles on the subject:


A Tutorial on Linear and Differential Cryptanalysis
A stochastic model for differential side channel cryptanalysis


Not top quality links, and especially not as practice-oriented as you seem to be looking for, but I think these last two could give you a bit of grip on the subject.