There's a large variety of violin cases available on the market but generally the more expensive the violin, the better quality protection you'll be looking for.
A beginner violin outfit will usually come with a shaped case which is will give a reasonable amount of protection to an instrument. However these cases are usually not very tough, and it's unlikely you'd want to carry a very expensive violin around in one on a regular basis.
You also have to be careful with cases that come with beginner violin outfits, as sometimes the straps on the back can be quite weak. This isn't usually an issue whilst the instruments are still new, but if any of the hooks come undone or bent out of shape, then there's a small risk the violin could fall off your back whilst you're walking around.
Rectangular cases usually offer more storage space than shaped cases. Some of them will also be heavier and it's worth making sure that you try and find out how heavy each case is before buying one, otherwise you might end up with something that really hurts your back or your shoulders.
Rectangular cases are often made with harder materials, and offer better protection to violins. So you'll generally find that people who have expensive violins will carry them in a rectangular rather than a shaped case.
Travel cases are also available, and these are very similar to normal violin cases except that they're designed specifically for travel - so sometimes they are unusual shapes, extremely light, or made of unusual material. If you take a violin in an aeroplane you can have all sorts of problems if you don't prepare correctly for the journey. A shaped case can often help you to keep within airline regulations whilst still making sure that you have adequate protection for the instrument. Although - make sure that you NEVER put a violin in an aircraft's hold!