Bcrypt has the best kind of repute that can be achieved for a cryptographic algorithm: it has been around for quite some time, used quite widely, "attracted attention", and yet remains unbroken to date.
Why bcrypt is somewhat better than PBKDF2
If you look at the situation in details, you can actually see some points where bcrypt is better than, say, PBKDF2. Bcrypt is a password hashing function which aims at being slow. To be precise, we want the password hashing function to be as slow as possible for the attacker while not being intolerably slow for the honest systems. Since "honest systems" tend to use off-the-shelf generic hardware (i.e. "a PC") which are also available to the attacker, the best that we can hope for is to make password hashing N times slower for both the attacker and for us. We then adjust N so as not to exceed our resources (foremost of which being the user's patience, which is really limited).
What we want to avoid is that an attacker might use some non-PC hardware which would allow him to suffer less than us from the extra work implied by bcrypt or PBKDF2. In particular, an industrious attacker may want to use a GPU or a FPGA. SHA-256, for instance, can be very efficiently implemented on a GPU, since it uses only 32-bit logic and arithmetic operations that GPU are very good at. Hence, an attacker with 500$ worth of GPU will be able to "try" many more passwords per hour than what he could do with 500$ worth of PC (the ratio depends on the type of GPU, but a 10x or 20x ratio would be typical).
Bcrypt happens to heavily rely on accesses to a table which is constantly altered throughout the algorithm execution. This is very fast on a PC, much less so on a GPU, where memory is shared and all cores compete for control of the internal memory bus. Thus, the boost that an attacker can get from using GPU is quite reduced, compared to what the attacker gets with PBKDF2 or similar designs.
The designers of bcrypt were quite aware of the issue, which is why they designed bcrypt out of the block cipher Blowfish and not a SHA-* function. They note in their article the following:
That means one should make any password function as efficient as possible for the setting in which it will operate. The designers of crypt failed to do this. They based crypt on DES, a particularly inefficient algorithm to implement in software because of many bit transpositions. They discounted hardware attacks, in part because crypt cannot be calculated with stock DES hardware. Unfortunately, Biham later discovered a software technique known as bitslicing that eliminates the cost of bit transpositions in computing many simultaneous DES encryptions. While bitslicing won't help anyone log in faster, it offers a staggering speedup to brute force password searches.
which shows that the hardware and the way it can be used is important. Even with the same PC as the honest system, an attacker can use bitslicing to try several passwords in parallel and get a boost out of it, because the attacker has several passwords to try, while the honest system has only one at a time.
Why bcrypt is not optimally secure
The bcrypt authors were working in 1999. At that time, the threat was custom ASIC with very low gate counts. Times have changed; now, the sophisticated attacker will use big FPGA, and the newer models (e.g. the Virtex from Xilinx) have embedded RAM blocks, which allow them to implement Blowfish and bcrypt very efficiently. Bcrypt needs only 4 kB of fast RAM. While bcrypt does a decent job at making life difficult for a GPU-enhanced attacker, it does little against a FPGA-wielding attacker.
This prompted Colin Percival to invent scrypt in 2009; this is a bcrypt-like function which requires much more RAM. This is still a new design (only two years) and nowhere nearly as widespread as bcrypt; I deem it too new to be recommended on a general basis. But its career should be followed.
(Edit: scrypt turned out to not to fully live up to its promises. Basically, it is good for what it was designed to do, i.e. protect the encryption key for the main hard disk of a computer: this is a usage context where the hashing can use hundreds of megabytes of RAM and several seconds worth of CPU. For a busy server that authenticates incoming requests, the CPU budget is much lower, because the server needs to be able to serve several concurrent requests at once, and not slow down to a crawl under occasional peak loads; but when scrypt uses less CPU, it also uses less RAM, this is part of how the function is internally defined. When the hash computation must complete within a few milliseconds of work, the used RAM amount is so low that scrypt becomes, technically, weaker than bcrypt.)
What NIST recommends
NIST has issued Special Publication SP 800-132 on the subject of storing hashed passwords. Basically they recommend PBKDF2. This does not mean that they deem bcrypt insecure; they say nothing at all about bcrypt. It just means that NIST deems PBKDF2 "secure enough" (and it certainly is much better than a simple hash !). Also, NIST is an administrative organization, so they are bound to just love anything which builds on already "Approved" algorithms like SHA-256. On the other hand, bcrypt comes from Blowfish which has never received any kind of NIST blessing (or curse).
While I recommend bcrypt, I still follow NIST in that if you implement PBKDF2 and use it properly (with a "high" iteration count), then it is quite probable that password storage is no longer the worst of your security issues.