What sort of lock do I need?
That looks like a Euro cylinder lock. You can remove it by undoing one bolt in the edge of the door then insert the key, turn an eighth of a turn and pull the cylinder out. Then measure the lengths in the diagram below. Or just measure externally.



You should get an anti-snap lock which means you need to specify which of the measurements applies to the outside facing end.

P.S. The outside end of the lock cylinder should not project from the surface of the surrounding faceplate, get a shorter one if you can (though they are only available in 5mm increments). This affects security.

P.P.S. Locks vary a lot in price but this lock is protecting all your belongings so it is probably worth spending extra on a good lock. Look for the TS007 3 Star standard and kitemark.





Top: 35/35 British Oval cylinder with thumb-turn (see also Scandinavian oval cylinders)
Middle: 50/40 Euro-profile cylinder (AKA Euro-cylinder)
Bottom: 0/35 Euro-profile half-cylinder

Features: 


The threaded hole is for the retaining screw.
The middle lock has the cam (actuator) in-line with the hull, in this position the whole cylinder can be pulled out (if the screw has been removed).
There are blanking plugs in the holes which contain the key-pins, driver-pins and springs. These particular cylinders are not designed to be re-keyable.
The reason I have these loose and not in use is because they have no anti-snap features and are mostly trivial to pick also.