Sorry for the... low quality map, I was looking for one with all the regions divided up and this was all I could really find for the moment. I also think the "Metro" area is too big (Caguas is in it! Seriously?).
Anyways, a brief introduction to each...
"Metro" is (obviously) San Juan and the surrounding municipalities ("municipios"... it doesn't quite roll off the tongue in English but it's not so bad in Spanish). Big cities besides SJ include (well, at least in this map) Carolina, which is where the Luis Muñoz Marín International Airport and Isla Verde are located, Río Piedras, where the University of Puerto Rico's main campus is, and apparently Caguas. I'm not really sure how to explain Caguas except that they have a web site talking about how great they are even though Caguas is in the middle of nowhere. I understand that they have a lot of industry going on there. I'm not sure. The only time I was in Caguas it was about 1 in the morning on Three Kings' Day (to be explained later) and we drove completely out of our way to try to see some house that supposedly was decorated with Christmas lights to some kind of record-breaking extreme. Of course, by the time we got there, there was nothing to see. I wish I were making that story up.
"Este" (East) includes cities like Loíza, the center of Black culture for the island, Luquillo, which has a nice beach (haven't been but I've seen it), Fajardo and Puerto Rico's smaller islands, Culebra and Vieques, and Humacao. There's also the famous park el Yunque here (in the Río Grande municipality).
"Sur" (South) is known for Ponce, the second largest city in Puerto Rico after San Juan. It's drier and hotter here in comparison to the rest of the island.
"Centro" (Center) is mostly mountains. It includes Utuado, Lares, where Puerto Rico's failed revolution, el Grito de Lares (the Shout of Lares), took place, and a few other smaller cities like Jayuya and Aibonito. It still hangs onto its Taíno and jíbaro roots.
"Oeste" (West) is a popular spot for tourists, particularly in Rincón, a gringo enclave and a surfing spot. It also has Mayagüez, a fairly large and pretty city, Cabo Rojo, which has a lot of beaches and the "Los Morrillos" lighthouse, and Aguadilla, which has some more nice beaches.
"Norte" (North) has Arecibo with its famous observatory and the world's largest radio telescope.
Keep in mind that this is just a very rough guide to the major cities. Puerto Rico, with a population of 4 million people in a tiny island, has a lot of interesting suburbs and rural areas that should be seen as well.