A router allows multiple devices to access one point of connection (such as an ADSL connection if the router's an ADSL router with a built-in ADSL modem, an uplink to the Internet if being used on a network, or another computer being used as an upstream proxy server). Routers commonly have DHCP servers installed, to provide connected devices with an INTERNAL ONLY IP address on a private subnet range. This means that whilst the internet will see only one requests from one IP address (the outward IP address of the router's ethernet adaptor / adsl modem), the router's routing table, combined with the abillities of the DHCP server, can cope with multiple requests to various sources (web sites, gaming servers etc), routed through the one external connection, but separated out and correctly distributed to the various connected devices.
An access point is a wireless version of a switch - it's a much simpler device, similar in function to a router, but with no DHCP server - it merely acts as a pass-through for multiple devices to transparently connect to (and share) a single uplink or other connection. As switches contain no DHCP server, they can be connected to a router to provide a) extra connectivity, and b) further network availability reach for wireless devices.