Type M copper pipe - is approved for residential construction (houses). It's the thinnest SUPPLY piping you can get. DWV (Drain, Waste and Vent) is thinner wall still, but it's only available in sizes 1 1/4 inch diameter and up, and it's used for drains and vents where there really is negligible pressure inside the pipe.
Here in Canada, Type M pipe will have red printing on it so that the inspector can tell what was used without cutting the pipe open. DWV piping will have yellow printing on it.
Type L - is thicker wall than M, and is approved for commercial use (office buildings and apartment blocks, etc.) Type L will have blue printing on it.
Type K - is the thickest supply pipe available, and it's used for underground applications. It'll have Green printing on it.
All of them have a 5/8 inch OD. It's the ID that changes. So, you can use the same fittings on Type K that you do on Type M or L. They don't make copper fittings in different grades like they do pipe.
It's actually hard to tell the difference between them by looking at the wall thickness in that they all look about the same. However, as mentioned by someone else, it's actually easy to tell by weight. Type L will weight more than you expect if you're used to Type M. Similarily, Type M will seem awfully light if you're used to Type L.
I try to avoid using Type M because it's been my experience (or bad luck) that Type M tends to go out of round much more easily when it's hot and you're fighting with a stubborn solder joint. When a copper pipe goes out of round when it's hot, it's a real problem because then you can't solder a fitting onto or off of it without a fight.