Watch this video, then try the exercises below.

IN SUMMARY

Sharp signs (#) tell you to play a note one semitone higher (to the right on the keyboard)

Flat signs (b) tell you to play a note one semitone lower (to the left on a keyboard)

Natural Signs 'cancel' sharp and flat signs (more on this later...)

These signs are collectively known as 'accidentals'. Accidentals are written on the stave in front of the note they are effecting.

If a sharp or a flat is written in front of a note, it effects every note written on that line for the rest of the bar, unless it is cancelled by a natural sign. In this example, there are three 'G's, numbered 1, 2 and 3. The first G has been sharpened, but the second G should also be sharp, because this sharp is still in effect. The composer will need to write in the natural sign if they want the third G to not be sharpened.

However, this effect only lasts until the next barline. The sharp sign in front of the note C (labelled with the number 4) is cancelled by the bar line, and the following note (5) would be played as a C natural.

The second important way in which sharp and flat symbols can have a lasting effect is key signatures. A key signature tells the musician which particular major or minor scale (more on this later) the music is based on. Key signatures usually  appear at the start of a line of music, (between the Clef and the time signature) and effect the whole piece of music (unless replaced by another key signature, or temporarily cancelled by accidentals written in the music...) 

The key signature in this picture is the key of D Major. The key signature tells the musician that ALL Fs and Cs should be sharpened, (unless otherwise indicated)