There is a very well-defined concept -- its called informational entropy. While there is a relationship between entropy and compressibility, don't think of entropy as a synonym for compressibility.
Entropy is the number of 50/50 decisions (like a flip of a fair coin) needed to make a password and we call each of these 50-50 decisions a bit of entropy. If you choose a password at random from a 1024-long list of common passwords, your password has 10-bits of entropy as 210 = 1024. If you choose a password by picking 12 random letters (taken from upper-case, lower-case, and numbers - for a total of 62 characters), we know there are 6212 such passwords which involved log2 (6212) ~ 71.450 ~ 71 bits (note 271.450 ~ 6212).
Informational entropy relates to a lower-limit on how approximately long it would take the best super-sophisticated attacker to construct your password. If they knew the exact model you used to create your password (e.g., they've seen other passwords you've generated or you've leaked your method), but your password generation method required ~71 random 50/50 decisions that means they'll have to try roughly 271 ~ 2,361,183,241,434,822,606,848 different passwords (so at a billion passwords a second it would take them tens of thousands of years to break). If they don't know your model, they'll have to try many more than that. (Because they first spent time brute-forcing 6-character passwords, going through a leaked password list of 100 million common passwords, then tried 7-character passwords, etc.
Note if an attacker also tried 8,9,10,11 character passwords first it doesn't make that much of a difference to the entropy if they eventually need to try 12 character passwords. That is 6212 ~ 271.450, but 628 + 629 + 6210 + 6211 + 6212 ~ 271.474 -- thus if your effort could crack passwords up to 12 characters long it could also crack passwords that are shorter with effort thats insignificant compared to cracking the 12-character password.
If your password is not based on random things, but say based on your spouse/children/pet's names or some commonly available fact with slight substitutions then it will be much weaker. The entropy calculation could be start from number of facts about you before someone would come up with the basis of the password plus number of random alterations (similar to http://xkcd.com/938 ).