Fingering and positions
The fingers of the left hand are conventionally named first (index finger) to fourth (little finger). When playing notes other than open strings (G, A, D and E), these fingers must press down hard, so that the string is shortened convincingly for a higher pitch. The standard intervals taught to beginners is tone, tone, semitone, tone (ie. G-A-B-C-D, D-E-F#-G-A, A-B-C#-D-E, and E-F#-G#-A-B). Of course the notes in between can be played by rearranging the hand position.
This is known as first position, where the first finger plays up to a tone above the open string. The next position usually taught is third position, where the first finger plays the note a perfect fourth above the open string (so, for example, third position on the A string would start on the D). All positions from first up to anywhere around tenth can be thus played, and two octaves on one string are considered fairly standard.