I'm a rank amateur, but here's an (oversimplified) back of the envelope summary:
The API ratings beginning with an "S" like SJ and SL refer to ratings approved for new US-market cars with gasoline.
BTW, "SL" is the most current of these.
API ratings beginning with "C" like CI-4 are ratings intended for US-market diesel engines.
ILSAC ratings are oil ratings put out by the Japanese manufacturers. GF-3 is the most current of these, although GF-4 will become the most current very soon now.
Oh, and you didn't ask, but the Euro carmakers have specs such as ACEA A1, A2 and A3 for gas engine vehicles.
Also B1, B2, and B3 for light diesel engine vehicles.
I think there's an ACEA "E" series for heavy diesel, but since I don't own any tractor trailers or farm tractors I don't follow that.
The different groups, such as group 1, group 2, group 3, etc indicate degree of "syntheticness".
Group IV and V oils are generally what EVERYONE will agree is synthetic.
Group III is controversial. Some people (notably Castrol in the past) market Group III oils as synthetic, and that is highly controversial in some quarters.
Group II is "highly refined" motor oil, but almost no one calls it synthetic.
Group I is... less highly refined than anything else, and nearly as far as you can get from synthetic without just pouring crude oil in your crankcase.
BTW, this is a beginner's view, and probably flat-out wrong in spots, but a good starting point.