Need for Speed: Most Wanted - For an arcade racer, this is the best of the bunch. They stick you in a relatively large sandbox area and you can just drive around for a while. Races are enclosed but in-between races you're free to go wherever you want. Police chases are fun because they're not enclosed and you can go from one end of the sandbox to the other trying to lose them.
Need for Speed: Carbon - More or less the same as Most Wanted although in my opinion the city they offer here isn't nearly as interesting as the one in Most Wanted.
Need for Speed Underground 2 - Interesting city to race around and they have interesting races where you flash your lights at another racer and then you have to race to get as far ahead of the other guy as you can. The leader makes the coarse by essentially going wherever they want to go and the follower has to try to overtake and become the leader. Otherwise, very similar to the other more recent NFS games aside from ProStreet which is strictly track racing.
Test Drive Unlimited - This one has a little bit more stuff than the NFS games. You have the entire island of Oahu to race around, not just one city. There's also little missions here and there such as picking up passengers, etc. You can also buy more properties with your race winnings. Although considered an arcade-ish game and not a sim, I felt that the driving model was a bit more stiff than the NFS games. You have to be far more careful around turns or bumps than the other games I mentioned.
Depending on your rig, the games will run from best to worst in this order: NFSU2, NFS:MW, NFS:C, and TDU. They're all perfectly playable on my old rig which was an A64 3000+ with 6800GT but more and more details need to be lost in order to get them to run ok. You might have to drop the res on TDU, depending on how fast you like your refresh rate.