Yes, there are still region codes ("zones") for DVDs. Common computer operating systems (Windows/OS X) usually allow you to change the region code of the DVD player only a few times over the lifetime of your hardware. The player will generally only play DVDs for which the region code matches that of the player.1
If you have a collection of region 2 DVDs that you currently play on your computer, those will still work after you move to region 1. If you purchase a new DVD in region 1, it won't play on your computer. In general, only region 1 DVDs are available in North America.
If you purchase a DVD player in region 1, it will probably only play region 1 DVDs. It may be possible to find a "region-free" player, usually they're off-brand players that can be modified to ignore the region code on played DVDs.
The region restrictions are, in my experience, most strongly enforced in region 1 (US/Canada). Here in New Zealand, consumers are expressly permitted by law to circumvent the DVD region code system. So most players you can buy here have a way of unlocking the region code functionality so it can play DVDs from any region. The first DVD player I bought here came with instructions to do so; the BluRay player I bought more recently needed some googling to find the magic code.
1. You can probably circumvent this restriction using freely available software.