If your bike is exactly the same as the one in the picture you linked to, then the frame and fork are disc compatible. To double check compare your bike to these pictures:
You'll need a mounting point on the fork, which will either be a post mount
like this, or an IS mount
like this. Both will work. You will also need a mount for the rear on your frame. It will be an IS mount and look something
like this, although it may be positioned inside the rear triangle.
As for wheels, they will need to be disc compatible also. They need to have the six mounting holes to fit the actual disc to the hub. These can be seen
here. If your bike does not have these, you will need to either get new hubs laced on to your existing rims, or purchase a new disc compatible wheelset.
When it comes to picking the actual brakes, you have two real options: hydraulic or cable actuated. Cable will be cheaper, but will require more adjustment. I'd advise against going for cheap cable disc brakes, as they tend to be pretty naff.
Avid BB7's are a good choice. BB5's will be slightly cheaper, but most people recommend spending the extra cash and going for the model up.
If your bike currently has an all-in-one shifter and brake lever setup (i.e. they are one complete unit on the bar with a single clamp,
as seen here) then going for cable discs will most likely be cheaper, as you can use the existing shifter brake lever setup, and hence not have to replace your shifters.
The other option is hydraulic discs. These require less maintenance, and tend to provide more stopping power and better modulation. If your bike currently has all-in-one shifter and brake levers, then you'll have to replace your shifters in order to use the hydraulic brake levers, adding extra cost on to the already more expensive hydraulic disc brakes.
If you don't have the mounts for the disc or calipers, then it'll probably end up costing a fair bit in order to replace the fork/hubs to make them disc compatible (could reach the £150 mark easily). If that's the case you'll have to decide whether it's worth spending that amount of money on the bike to you. If not, then it may be time to start saving for a dedicated mountain bike!
If your frame doesn't have the mounts, but the fork does, you could always put discs on the front end only.