Makes a spherical world flat! Basically split the globe up into segments (like an orange) and then fit the outside face of that segment to a rectangle. All done by horrible maths.
Much easier to work out distances between points (as it's metric) than Geographical coordinates (in which a distance would be in arc seconds).
Not used too much in the UK since we have the British National Grid, this is a transverse mercator projection so not too different but means we don't have to cross utm zone boundaries as we move across the country.
EDIT the zone number essentially refers to the segment of the globe that you are in (each segment is 6 degrees wide). it does indeed make as massive difference to your coordinates since each zone has it's own origin (0,0 in eastings and Northings) so coordinate positions are only relevant to the zone you are in.