I have some pretty big sync's to do, but I can't sit around my computer making sure it's awake and unlocked all day/night.
Does Dropbox sync when the computer is "Locked" or asleep?
We don't know what your system is, so I'm a little fuzzy on the definition of 'locked' for you.
Under a Windows OS, if you lock the screen (that is, stay logged in, use Win+L or 'lock' under the start menu to lock it), then yes, Dropbox will continue to run and sync.
Under a Windows OS, 'sleep' means that the computer is off and not running programs, but memory is being held live so that it can wake up very quickly. In this state, NO, dropbox can't sync, because it's not running.
Going back to the original subject on top of @Michael Kohne's answer, there seems to be a pitfall on Mac in terms of lock; in default setting (I'm on OSX 10.9.5), OS might go to sleep during lock. To disable that feature, see this post.
With my Dropbox on Mac, I found by looking at my Recent Event page on dropbox.com that the sync only happens while I'm at the machine, although I didn't let it sleep before I left (to sync the huge batch of files). The solution above worked for me -- now it syncs while I'm away.