It's a pretty blurry thing, I believe. You are already 'analyzing the blockchain' when you're following some coins you sent to new recipients. You could also count the number of addresses that have more than 1 BTC in them. But you could also go and use sophisticated algorithms in order to find out certain distributions or determine to what probability some coins are connected to a certain address.
Neat, this seems like kinda of a decryption tactic, where you analyze letter frequency to break the encryption. Is this something similar in the block chain, where you analyze transaction frequencies to find irregularities?
Umm, well it depends on what you're trying to accomplish with your analysis, I guess... Analysing patterns makes only sense if you're looking for something unknown or try to find something you suspect is there, but don't quite know how to identify or quantify it. Other analysis may just consist of following the addresses coins "travel"...