This simple Bourree was written by J.S. Bach (1685-1750) and is currently a grade 2 classical guitar exam piece for the ABRSM.
Bach was undeniably one of the greatest composers of all time, but unfortunately the classical guitar as we know it didn't exist in Bach's day, so he never wrote any music for the instrument. He did, however, write music for its close relative, the lute, (he was a friend of the great lutenist and composer Silvius Leopold Weiss), and these lute pieces adapt well for the guitar.
In addition to the lute works, many of Bach's compositions for other instruments also work well on the guitar. In particular he wrote a number of sonatas and partitas for solo violin, and six suites for solo cello, all of which can be adapted ("transcribed") to play on the guitar.
This Bourree is one of the six movements from Bach's 4th cello suite. In fact it is only one half of a movement, as this is the second of a pair of Bourrees that make up the 5th movement in the suite. The whole suite was originally written in the key of Eb major, but the guitar transcription shown above is in the key of A major, as this key works better for the guitar.
You can see that the Bourree is written with two different parts or 'voices' playing at the same time: an upper melody line with the note stems pointing upwards, and a bass line with stems downwards. Bach was a master of this type of music - known as counterpoint or contrapuntal music - and he could write and even improvise (make up on the spot) music with as many as six different parts all playing at once!
With this kind of music I always recommend practicing the piece by playing each of the lines separately to get to know each individual part. Then when you play them together again it becomes easier to bring out the different voices. Aim to create the illusion that there are two guitars playing at once.
Notice how the music starts with an incomplete bar with just one crotchet beat. This means that the musical phrases in the piece don't end on the 4th beat of the bar, but on the 3rd beat. The 4th beat then becomes the start of the next new phrase. This feature is characteristic of the Bourree, which was originally a type of French dance.
Please note that the guitar MP3 recording doesn't include the last 8 bars of the music (bars 17-24) as these bars are an exact repeat of the previous 8 bars (bars 9-16).
Bach wrote a number of different Bourrees, so you may find it interesting to compare this one with some of the others to get an idea of what they have in common. Here's a link to the Bourree from the Lute Suite in E minor, BWV996, which is a particular favourite amongst guitarists. It's a harder piece as it's a grade 6 exam piece, but there is a soundfile to accompany it so you can hear how it sounds even if it's too difficult for you to play at present.
Finally, one other suggestion I can offer is to try and hear a recording of this A major Bourree (or even better, the whole suite) played on a cello. You'll probably hear the piece in a very different way from listening to the guitar version, and it may give you some fresh ideas for playing the music.