It is pretty simple: if everything works, no malfunctions, no false signals, then the aircraft is in normal law. You have all the stability and protections that Airbus intended the aicraft to be operated with. In real life, the aircraft will be in this state for 99.999% of the time. In my 12 years of flying, I have never encountered alternate or direct law in line flying. Only in simulator training.
If certain things go wrong (malfunctions, detected wrong signals), the flight control computers cannot function properly in normal law. There may be important signals missing to compute the proper reactions to stick inputs or environmental disturbances. For example, if you lose some of the speed signals, or lose flight control surfaces (hydraulic actuation or electric control), the aicraft reverts to a simpler flight law.
You have less protections in alternate law, and no protections at all in direct law. The aircraft reverts to this state, because it cannot guarantee that the protections will work the way they were designed, and they could even do harm (if all speed indications are lost or wrong, a low or high speed protection would make the aircraft unflyable).
Some Airbus aircraft revert to direct law with gear down (A320), and some do not (A330). With direct law, you have -- as the name suggests -- direct stick to surface control. Sometimes, the aircraft goes straight into direct law (loss of all ADRs for example).
There are even more flight control laws, that are not indicated to the pilots. You have flare law (when below 50 ft radio altitude for landing), ground law, abnormal attitude law, mechanical backup...
To sum it up: Airbus has implemented flight control laws other than normal law to cope with malfunctions and abnormal situations.