As I understand it, full disk encryption encrypts all data on a HDD and protects that data with a password. If I turn my PC off and it gets stolen, all my data, passwords etc. are protected. But I always leave my PC on and put it to sleep when I'm not using it. Should I still consider encrypting my disk?
Bear in mind that full-disk encryption will slow I/O, and that the keys must be stored in RAM while the machine is on. If you leave it on, not only is the data available for immediate access, but if you aren't set up carefully someone might be able to recover the keys before taking the machine away, in which case your encryption won't mean a thing.
On the other hand, sleep mode would fail as soon as somebody unplugged the computer, assuming a desktop, so you might get a degree of security from that.
The key thing would be not to regard the disk encryption as rendering your machine bulletproof. It's certainly a security increase over no encryption at all, but (and I would say this even if you were in the habit of powering the machine down) it's not a perfect solution, so you should still be cautious if you anticipate a need for security (which, of course, there always is, at least to some extent).
One option to consider, assuming your disk-encryption solution supports it, would be to accept a hit to your time-to-desktop, and use hibernation (suspend-to-disk) instead of sleep (suspend-to-RAM).
If this is a personal computer, there's a very high chance if it's stolen they'll be unplugging the machine, at which case it gets powered down. Most break-ins aren't done by computer hackers waiting to find an encrypted machine turned on.