Accidental
A sign that raises, lowers, or restores a note.
What it means
Accidental (German Vorzeichen) is the umbrella term for the signs that change a natural note in its pitch or cancel a change. There are three basic forms: the sharp raises by a semitone, the flat lowers by a semitone, the natural sign restores the natural note. Alongside these stand the double sharp and double flat for a change by two semitones. With these signs, the twelve semitones of the system are obtained from the seven natural notes.
In the notation
Two positions are to be distinguished. At the beginning of the line, after the clef, the signs form the key signature: they apply to all octave registers and to the whole piece, until a new key signature replaces them; their order is fixed. Within the bar, before a single note, they stand as in-bar accidentals: they apply only to this octave register and only until the next barline. A sign in brackets is a courtesy sign, which changes nothing but serves as a reminder.
Musical use
The key signature does not name the key but only the note stock; the same key signature applies to a major key and its relative minor. In-bar accidentals accomplish everything beyond that: the raised leading tone in minor, the leading tones of secondary dominants, altered chord tones, chromatic passing notes. Why a particular sign is chosen and not the sonically equivalent other one is decided by function: the spelling is meant to show which degree is intended and where the note goes on. This is why an ascending line has a sharp and a descending one a flat, even though both hit the same key on the piano.
A distinction is made between key signatures at the beginning of every line, which apply to the whole piece, and in-bar accidentals before individual notes, which act until the end of the bar.
Performance practice
Say the key signature aloud to yourself before the first note and name the affected notes, not just their number. Within the bar, watch the two limits of an in-bar accidental — the barline and the octave register. And with chromatic passages, read the signs deliberately as signposts: a sharp mostly points upward, a flat downward.