This is right from Wikipedia - Replicator (Star Trek):
A replicator can create any inanimate matter, as long as the desired molecular structure is on file, but it cannot create antimatter, dilithium, latinum, or a living organism of any kind; in the case of living organisms, non-canon works such as the Star Trek: the Next Generation Technical Manual state that, though the replicators use a form of transporter technology, it's at such a low resolution that creating living tissue is a physical impossibility.
- Latinum became the de-facto currency because it could not be replicated.
- I don't think dilithium's un-replicatability has ever been explicitly stated, but dilithium mines wouldn't be necessary if it could be replicated.
- Likewise for antimatter, I don't recall any reason it can't be replicated. The most likely reason it doesn't get replicated is because it's simply inefficient to do so - the energy used to replicate it could instead have been used directly for whatever the antimatter reaction would have been used for.
- There are three instances where the replicators and transporters (replicators being an offshoot of transporter technology) have arguably created "life". Because it borders on impossible, though, it's much easier to say "can't be done" - Two of those three were life forms due to accidents, and have never been successfully duplicated, while the last one was simply biological matter:
- Thomas Riker, during a transporter accident, as discovered in TNG 6x24, Second Chances. This is likely dangerous to even attempt to duplicate, and raises too many ethical concerns of copying a person.
- The Enterprise-D's offspring, in TNG 7x23, Emergence, which AFAIK the Federation has never figured out how it happened, nor has been able to duplicate it. This "life" would not have been recognizable as such without the scanners anyway, and required continuous transporter/replicator use for hours, making it extremely difficult to duplicate.
- Worf's new spine, in TNG 5x16, Ethics, was created with a Genitronic Replicator, a brand-new invention that had to scan the genetic information of an already-existing organ in order to duplicate it. This was not life, exactly, but it is biological replication.
Additional note that may clarify the "life" part. In a comment, DaveP asked:
One thing that hasn't been touched on here is food. While replicated food (and not just tea, Earl Grey, hot) isn't necessarily biological, it is surely organic. Or is it?
It's not quite that simple. Occasionally in Sick Bay, you'll hear someone say "... at the cellular level", or "... at the molecular level". There's at least one level further, the "quantum level", and that's the level that's an issue.
Keep in mind, this is mostly theory picked up from all across Star Trek.
Transporters are capable of scanning/storing/transmitting/rematerializing matter across all these levels, otherwise transporters would just leave you a vegetable, unable to think. However, the quantum level (the level needed to deal with neural patterns) takes an absurd amount of memory to actually store it long-term.
Because of this, replicators are made to only work at the molecular level, so the stored patterns aren't too massive. This works for tools, food, and and so on, since most people can't tell that food replicated this way is any different from non-replicated food (although some claim to be able to tell the difference..), so there's no need to give replicators any more resolution.
This would be why a special type of just-invented replicator was necessary for Worf's spine - bodies (and body parts) in general are too complex for replicators to handle.