Are specific private keys weaker than others? I've heard some discussion about how certain private keys can be considered weaker because they were generated using a weak randomness algorithm. I have very little experience in this subject and currently most of my bitcoins are tied to a private key that I generated a while ago, using MultiBit V0.5.14. 

I'm wondering whether anybody with more experience about this can explain this subject more in detail.
They are weaker in the sense that they are easier to guess because the attacker has a more limited set of private keys to guess from. All private keys are equal in terms of just taking a public key and trying to guess the corresponding private keys, because, without extra information, one has to check every possible private key (until the correct one is found). But when one knows the private keys weren't truly random, by implication one knows that there is a certain pattern to the possible private keys. This means that the attacker can guess private keys from this pattern which is smaller/quicker than guessing all possible private keys. I can't speak to the specific weak generator in question, as I haven't heard of it, but with a general understanding of cryptography, I can say this is what people mean when they speak of weaknesses in the key generator.