3G is a cell phone signal which is data and is around 2mbps at it's peak speed. Pretty good, it keeps up with watching YouTube, pandora, and emailing. 4g is a new technology which is LTE and HSPA+. 4g has speeds of up to 10mbps, screaming fast! This means that you can watch YouTube videos in absolute high quality even with just one bar of 4g. The advantage of 4g is that it is a more reliable signal because it is still just as fast with one bar as 3G is with full bars. The problem with 4g is that it is taking time for companies to make phones that are capable of these speeds and multitasking abilities. All smartphones are capable of 3G, but there are a few that are 4g ready. Sprint was the first cell phone company to have 4g phones. The HTC evo was the very first 4g phone in America.
You are correct, 3G and 4 G are basically 3rd Generation and 4th Generation, but the other meaning has to do with the network generation the phone is compatible with, not the generation of the phone itself. 1G would be the network that Zach Morris, from Saved By The Bell, would have connected to. Very slow and the concept of texting wasn't even thought of yet. As the networks grew and people kept demanding more usage out of their cell phones other than just being able to make a call, the networks grew. You can find out all the details here, it's really an interesting read.