What is the effect of the different AES key lengths? How does a changing key length affects the ciphertext, not only in case of AES, but in general? I know that the key spaces become much larger and the number of rounds in case of AES changes, but is security really that much enhanced that if I choose 256 bits as key length it is sort of unbreakable?

Something else what I'm thinking about currently, any idea if it is possible to get any details about key length or algorithm of an observed ciphertext? For example when measuring the entropy (or anything else) is there any difference e.g. between AES and DES?
For symmetric algorithms (like AES or DES or RC4 -- but not for RSA or ECDSA), a key is a sequence of bits, such that any sequence of bits of the same size is a potential key. So longer keys means more possible keys.

Exhaustive search is about trying all possible keys until a match is found. It is an absolute limit to the strength of an algorithm: exhaustive search is always applicable, there is nothing an algorithm can do against it. The only defence against exhaustive search is to expand the space of possible keys, i.e. have longer keys. Fortunately, this is very effective: the largest exhaustive search experiment which was ever reported to have been performed was for a 64-bit key (a search for a 72-bit key is underway, but not completed yet). Traditionally we use "80 bits" as the practical limit, but technology improvement may force us to increase that. Every additional bit doubles the hardness of exhaustive search. See this answer on security.SE for some additional discussion. Bottom line: 128 bits are safe.

It is somewhat delicate to compare strengths of 128-bit and 256-bit keys: while the latter is a much wider key space, the former is already in the "can't break it" zone, and you cannot get better than that, practically speaking. Thus, stating that AES-256 is stronger than AES-128 would be kind of abusive.

There are a few gotchas:


DES uses 64-bit keys (and 3DES uses 192-bit keys). However, the algorithm specification is such that DES totally ignores 8 of the 64 key bits (24 ignored bits for 3DES). Thus, the key size of DES is often said to be 56 bits (168 bits for 3DES).
AES morphs with the key size: with a longer key, there are more rounds. This is not about exhaustive key search; this is because a longer key is a promise of "higher academic strength" ("academic" means "we assume that the attacker can perform 300-bit exhaustive searches"). In an academic way, AES needs exhaustive search to be the best attack (i.e. attacks on the algorithm internal structure must be slower). A 256-bit key raises the bar: the non-exhaustive-search attacks must not be more effective than $2^{256}$ work factor. Hence the extra rounds. In practice, the extra rounds do not buy you more security (just like the larger key does not make the algorithm really less breakable than the already unbreakable AES-128); what the extra rounds do is that they make encryption 40% slower.
When an algorithm morphs with the key size, this is no longer "the same" algorithm, so anything goes. AES-256 could theoretically have weaknesses that AES-128 does not have (there is no guarantee that the extra rounds do not harm security).
Symmetric key size is very rarely the weak point of a system which uses cryptography. If you use 64-bit keys and that key size is indeed the best attack point, then you can consider yourself extremely lucky: the rest of the system must be inordinately robust.