A coil-over system is a system where the coil actually surrounds the shock. Nowadays in the import market they have taken on the meaning of height-adjustable coil-overs which allow you to adjust the height of the car. Now many folks think that this is so you can vary the height to your liking. Actually they are designed so you can adjust weight over each corner and balance the car using weight scales on each corner. They are NOT designed for you to adjust for winter/summer although you'll find a lot of kids use them for this purpose.
Now then, what are the advanatages apart from the ability to balance your chassis and set the frame height perfectly you ask? Well when going to a full coil-over suspension package you get the advantage of having a matched spring/shock setup and often times having a shorter shock casing to help increase travel (which you would otherwise lose with just a spring/shock combo). This allows the spring and shock to work their best since the dampening has been tuned somewhat for the spring rates used. Most of the time you have a nice adjustable strut to help dial in what you want out of the system. These nicer suspensions often increase the diamter of the shock so you get increased dampening capacity despite having a shorter case. Also, they are much more compact units than the stock springs (which look like huge truck springs). Lastly, on the more expensive units they are alluminum and reduce unsprung weight. Hope this helps.