To be less technical, a wireless router provides both router services (setting up a subnet) and wireless services (enabling wireless connectivity).
An access point provides only wireless services. The subnet is still managed by the router.
Before I moved, I used to have an access point, because I already had a perfectly good wired router when I got it.
The router address was 192.168.1.1, establishing a subnet of 192.168.1.x with a gateway address of 192.168.1.1.
I gave the access point an address of 192.168.1.50 (easy to remember), within the existing subnet. The gateway address for the subnet was still 192.168.1.1, because the router still managed the subnet and the DHCP; all the access point managed was wireless connections, making them possible and forwarding them via wire to the router, then routing incoming traffic for wireless connections to the wireless devices.
Here's a link to the best tutorial on subnetting I have found. It's in HTML 3.x format, but don't be misled by the lack of HTML 4.x/web 2.0 bells and whistles. It's well-designed, clearly written, and easy to understand. It used to exist on a dot-edu site; my guess is that the author retired and moved it to a personal website: http://www.ralphb.net/IPSubnet/index.html