The mandolin has eight strings arranged in four pairs of two (a pair of strings is known as a course). The stings are tuned G D A E (G being the lowest tone). This can be remembered by the rhyme Good Dogs Always Eat. You tune the mandolin by turning the tuning pegs in the appropriate direction until the string is in tune. In order to ensure the string is in tune you can use a chromatic tuner or tune by ear.
Chromatic Tuners can be bought from any good music shop and via the internet. There are ones that clip onto your mandolin. I would recommend these as they do not pick up any other noises in the room such as other instruments. They have a visual display that shows you when the strings are in tune. These are useful when playing with other people as it keeps everybody in perfect tune.
There also lots of tuner apps available for Apple and Android devices. These are good if you are in a room on your own. However if you are in a room with lots of other musicians then the app will pick up other intruments as well as your own leading to confusion!
To tune by ear you can use a pitch pipe or tune to another instrument like a piano ( click here to see what the notes are on piano) or accordion. Turn the tuning pegs until the two instruments sound in tune. Which direction you turn the tuning peg will depend on whether your strings are higher(sharp) or lower(flat) compared to the reference tone. One you have one course of strings in tune you can then tune the other strings to that course. For example if you have the second course (A strings) tuned then you would follow the following procedure to tune the rest of the strings.
Step 1: Play the 7th fret of the A strings. This gives the E note which should be the same as the E strings(1st course). Tune one of the E strings to the 7th fret of the A strings. Then tune the 2nd E string so both E strings are in unison.
Step 2: Play the 7th fret of the D strings. This gives an A note which should be the same as the A strings (2nd course). Tune one of the D strings so that when the 7th fret is played it is in tune with the A strings. Then tune the 2nd D string so both D strings are in unison.
Step 3: Play the 7th fret of the G strings. This gives a D note which should be the same as the D strings (3rd course). Tune one of the G strings so that when the 7th fret is played it is in tune with the D strings. Then tune the 2nd G string so both G strings are in unison.