Is AES-256 weaker than 192 and 128 bit versions? From a paper via Schneier on Security's Another AES Attack (emphasis mine):

In the case of AES-128, there is no known attack which is faster than the 2128 complexity of exhaustive search. However, AES-192 and AES-256 were recently shown to be breakable by attacks which require 2176 and 2119 time, respectively. While these complexities are much faster than exhaustive search, they are completely non-practical, and do not seem to pose any real threat to the security of AES-based systems.
In this paper we describe several attacks which can break with practical complexity variants of AES-256 whose number of rounds are comparable to that of AES-128.

Is the implication here that AES-192 is stronger than AES-256? If so, in simple terms, how is that possible?

The attack exploits the fact that the key schedule for 256-bit version is pretty lousy -- something we pointed out in our 2000 paper -- but doesn't extend to AES with a 128-bit key.

Again, in simple terms, what does that mean?
In practice, does this mean I should drop AES-256 in favor of its 128-bit counterpart, like Schneier recommends in the comments?

That being said, the key schedule for AES-256 is very poor. I would recommend that people use AES-128 and not AES-256.
AES is an algorithm which is split into several internal rounds, and each round needs a specific 128-bit subkey (and an extra subkey is needed at the end). In an ideal world, the 11/13/15 subkeys would be generated from a strong, cryptographically secure PRNG, itself seeded with "the" key.

However, this world is not ideal, and the subkeys are generated through a process called the key schedule, which is very fast but not a decent PRNG at all; it is meant to offer sufficient security in the very specific context of producing subkeys for the AES only. The key schedule of AES was already known to be somewhat weak in some ways, allowing some exploitable structure to leak from one subkey to another, and this means related-key attacks.

Related-key attacks are not a problem when the encryption algorithm is used for encryption, because they work only when the victim uses several distinct keys, such that the differences (bitwise XOR) between the keys are known to the attacker and follow a very definite pattern. This is not the kind of thing which often occurs in protocols where AES is used; correspondingly, resistance to related-key attacks was not a design criterion for the AES competition. Related-key attacks can be troublesome when we try to reuse the block cipher as a building block for something else, e.g. a hash function. In the formal land of academic cryptanalysis, related-key attacks still count as worthwhile results, despite their lack of applicability to most practical scenarios.

The key schedules for AES-128, AES-192 and AES-256 are necessarily distinct from each other, since they must work over master keys of distinct sizes and produce distinct numbers of subkeys. It turns out that the version of the key schedule for AES-128 seems quite stronger than the key schedule for AES-256 when considering resistance to related-key attacks. It is actually quite logical: to build a related-key attack, the cryptographer must have some fine control over the subkeys, preferably as independently from each other as possible. It seems natural that the longer the source master key, the more control over subkeys the cryptanalyst gets -- because the related-key attack model is a model where the attacker can somehow "choose" the keys (or at least the differences between the keys). In the extreme case of a 1408-bit master key which would simply be split into eleven 128-bit keys, the cryptanalyst would have all the independent control he could wish for. Therefore, an academic weakness relatively to related-key attacks should, generically, increase with the key size.

The apparent paradox of the decrease in academic security when the key size increases highlights the contrived nature of the related-key attack model.