Between the notes of the musical scale lie sharps and flats. The easiest way to visualize this is with a piano: the sharps and flats are the black keys, while the white keys are the natural notes. The key signature (a handful of symbols at the beginning of the music) tells you which black keys to hit. It's particularly easy on my harp, the traditional lever (Celtic style) harp, because the set the key, you flip up the levers.
So what is an accidental? An accidental is a sharp or flat that doesn't appear in the key signature, or a natural note where (again, according to the key signature) a sharp/flat should be.
The difficulty of this on a lever harp is that it requires flipping a lever, which means taking one hand away from playing. Not only that, the string is still vibrating, so if you return the lever to its previous position too soon, you get unwanted bonus sound. The only exception is if the accidental consistently appears throughout the music. In that case, you can set the rogue lever and leave it.
From a fiction perspective (since this is, after all, a writer's blog), who cares? Well, any portable harp will almost certainly be traditional style; the pedal style harps played in the orchestra require a framework of a size that makes casual carting-about prohibitive. Wire strung harps do not have levers, which means that the key has to be set by tuning the instrument before beginning play. For historical context, levers are a (relatively) recent invention, so nylon / gut strung harps may have the same limitations. But the actual technical design of levers isn't that complex, so it's possible for a typical fantasy world to have them.
What it boils down to is there are certain songs that are impossible to play on certain harps. The big joke at my luthier (yes, harp players and harp makers have geeky in-jokes) is that a cross-strung harp is the only harp on which you can play "Flight of the Bumblebee."
Of course, the biggest harp joke is: "How long does it take to tune a harp? Nobody knows."
(Because by the time you finish, it's out again ...)
A pet peeve that I've seen in stories is strings that break from play. Nope. The amount of pressure being applied upon each string is far more than the average human's ability to pull. The most likely time for strings to break is while tuning (we've all done that, believe me) or with a sudden drop in temperature. Wood shrinks, increasing the continuous tension on the string and ... twang.