Accidentals are sharps or flats that are not in the key signature. They are added in next to the note so the player knows whether to sharpen the note or to flatten the note.

Sharps and Flats

If you sharpen a note, you play it up one semitone. If you flatten a note you play it down one semitone.

Naturals

The note, and to neutralise the sharp or flat in the key signature or in that bar.

Accidentals appear before a note on the stave, and tell you whether to sharpen the note, flatten the note or to play it as a natural. 

Accidentals also carry through the bar. If you have a sharp/flat on the note of beat one, and the same note on beat 4, that note will also be sharpened/flattened. However, in the next bar you will have to put in the accidental again because it doesn't carry over from the previous bar. Similarly, the accidentals do not carry through octaves, if you had a note with a sharp or flat in one octave, you would have to put a sharp/flat another octave if that is being used.

On the right, there is a sharp, flat and a natural sign. These are the three accidentals you will find in music. Below it, you can see there is a piano keyboard. If you look at D, you can see that D# goes up one semitone (to a black note, Db goes down one semitone (to another black note) and D natural stays as it is.