It's a descendant of the folk-plays once widespread in Britain. Within communities, groups of men (sometimes boys) would join together to celebrate some important date in the year by acting out a play appropriate to the time, in the streets or inns, or even private houses. They were usually a way to gather contributions, in cash or kind, to the welfare of "the labouring classes".
What were these important dates? They usually marked a change of season. The Christmastide mummers are sometimes claimed to be a relic of the Winter solstice festivities, celebrating the turn of the year when the days again start growing longer, ie the "death" and "resurrection" of the Sun. In January, there were Plough Plays, then Springtime (Easter), and Autumn (the Souling plays of All Souls Day, November 1st).
Why "Mummers"? Its just an old word for Actors or Players.
When did the Mummers Plays start? No-one knows, but the form we know seems to be derived from a small number of printed chapbooks which appeared about 250 years ago. Of course, the chapbooks were probably just the first collections of plays which had been in the oral tradition for many years. What is sure is that the "chapbook" plays returned to that tradition; they were handed down, with variations, from generation to generation of mummers.
Are they related to the Mystery Plays? No, they're purely secular, there was no Church connection. Also, the language and action is broader and (we hope) more comic. There's a touch of Shakespeare's "Rude Mechanicals" about the mummers.
Is there a common theme in the plays? Yes, many of the plays are described as "Hero-Combat", i.e. a Hero (St George) fights and conquers all manner of enemies (The Dragon, The Turk, Bold Slasher,etc). The other major motif is "death and resurrection", so inevitably the Hero's victims get "cured", and live to fight another day.
Why are some faces blacked-up? The convention used to be that the character in the play was NOT the person who played him, so the mummer's identity was disguised by make-up (often soot) or a mask or other obscuring costume. Naturally, everyone knew who was who, but even a player's mother would address him as his Character and not by his name!
Is it a male-only activity? Well, it used to be, when it was agricultural workers trying to supplement their income. Nowadays, anyone can be a mummer, so you might see some interesting cross-gender casting.
Are St Alban's the only Mummers around? By no means. For a comprehensive, searchable list, see Master Mummers, at http://www.mastermummers.org/groupslist.php or use your favourite search engine to find your nearest group.