A wireless router is a router that would allow you to connect without any wires or any other interface. There are two major types of wireless routers. One used for the specific wireless use, the other is an IP router with 802.11 interface card along with an antenna, the most common one. The purpose of the wireless routers is to provide Network Address Translation or (NAT) and they act as switches and firewalls also.
If you are searching for high speed transmission then you should be looking for 802.11g or 802.11a wireless router. But if you think that you can manage with average speed of 11Mbps and are looking to save money then 802.11b is your choice.
When you think that the range of your wireless router is to be increased then you should opt for changing the antenna alone. In the world of 802.11 Wi-Fi, many people would call wireless routers as the access points.