Wi-Fi teminology can be a little confusing because there are many devices that seem to do the same thing, but in difference in details.
Maybe theses definitions form Wikipedia will help see the differences.
Routers connect two or more logical subnets, which do not necessarily map one-to-one to the physical interfaces of the router. The term "layer 3 switch" often is used interchangeably with router, but switch is a general term without a rigorous technical definition. In marketing usage, it is generally optimized for Ethernet LAN interfaces and may not have other physical interface types. In comparison, a network hub does not do any routing, instead every packet it receives on one network line gets forwarded to all the other network lines.
WIRELESS ACCESS POINTS
In computer networking, a wireless access point (WAP or AP) is a device that allows wireless communication devices to connect to a wireless network using Wi-Fi, Bluetooth or related standards. The WAP usually connects to a wired network, and can relay data between the wireless devices (such as computers or printers) and wired devices on the network.
A wireless router is a network device that performs the functions of a router but also includes the functions of a wireless access point. It is commonly used to allow access to the Internet or a computer network without the need for a cabled connection. It can function in a wired LAN (local area network), a wireless only LAN, or a mixed wired/wireless network. Most current wireless routers have the following characteristics:
LAN ports, which function in the same manner as the ports of a network switch
A WAN port, to connect to a wider area network. The routing functions are filtered using this port. If it is not used, many functions of the router will be bypassed.
Wireless antennae. These allow connections from other wireless devices (NICs (network interface cards), wireless repeaters, wireless access points, and wireless bridges, for example).
Hotspots use AP's connected through ethernet routers that provide the dhcp and IP addy's THROUGHT the WAP's to the client devices. The ap's control interface should allow you to designate a router with internet access as the GATEWAY IP for the AP device.
Basically an AP IS a router, with ONLY wireless connection between it and the client devices.
Clear as mud??