BC (British Cycling, formerly the British Cycling Federation) operates a points-based system in competitive events. Each type of event that it is involved in (BMX, cyclo-cross, road & track, etc) is scored separately, and road racing is the one that gets most attention. There are several things you need to know about earning BC points, and BC road-racing points in particular, so read on.
The first thing you will need is a full racing licence. You can race just on a provisional licence, but you won't earn any points. You get the choice of racing licence when you buy your BC membership, and a full licence is an extra £32 per year (or so).
Once you have received your membership and licence pack, got a passport photo and assembled your ID card, you are ready to race! As a rookie road racer, you start off in the "4th cat", and at the end of each season your points tally determines whether you move up to 3rd cat. Note that, once you're 3rd cat, you can't get downgraded to 4th cat in subsequent years. In 2nd, 1st and Elite Cat, there's a minimum number of points you must earn to avoid that downgrade. Full details at the BC page: Road Categories (Riders) & Classifications (Events).
Both road races and track races contribute to the points system. The minimum grade for a road race is "Regional C+", and for track it is "Regional B"; for these races, the winner takes 10 points, and 10th place gets 1 point. If you want to earn points in a race, make sure that it qualifies; a "Regional C" race just won't earn you anything. The number of points that all of the different race classes award is at the BC page: LICENCE POINTS EXPLAINED - ROAD & CIRCUIT RACING.
A final note about keeping track of your points: go to your British Cycling Dashboard regularly! Not only can you find events to enter there (and even enter a lot of them online), you can also see your results and how many points you've earnt.