Broadband Routers normally combine a number of features of other networking devices. In addition to normal routing services, they normally provide a built-in network switch, firewall features and commonly a DHCP Server.
A standard router is designed to build routing tables through the use of a common routing protocol alongside peer routers, and use that routing table to determine which interface out of which to send an IP Packet or stream of IP Packets. When the router receives an Ethernet frame containing an IP packet in the payload, it checks the destination IP Address in the packet, performs a 'Logical And' process on the IP Address and assigned Network Mask. The result will provide the destination Network Address which the router looks for in the routing table. A router can have multiple interfaces that connect to different areas of an Internetwork, such as an Ethernet Interface, DSL interface, Cable Interface or Point-to-Point interface to name a few.
Some routers are designed to support only a few connections and deal with a relatively small amount of data traffic, whilst some ISP routers are very powerful devices that can deal with literally hundreds of millions of packets per second.
A Broadband router will normally have a DSL interface that connects through your phone line to your ISP, or a Cable Interface which connects via an Ethernet Cable through a Cable Modem and on to the ISP via a coaxial cable. Sometimes Broadband Routers are called Internet Connection Sharing because they allow a number of peripheral devices such as PCs to share a common DSL or Cable connection.
Most consumer Broadband Routers will provide an in-built 4-port Ethernet switch for connection to local devices by means of an Ethernet patch cable and a Wireless interface complying with one of the IEEE 802.11 standards such a IEEE 802.11g. This allows a number of computers to have a mobile capability and is ideal for such devices as Laptop Computers and SmartPhones with wireless capability. Both the wired and wireless interfaces will be served by a DHCP Server to dynamically allocate IP Addresses to connected devices. Additionally the router will support Network Address Translation (NAT) to allow all local devices to share a single IP Address when accessing the Internet.
Broadband Wireless Routers need an extra layer of security to protect the Wireless LAN from potential interlopers and hackers. Most modern Broadband Routers support a combination of security features such as WPA - WiFi Protected Access and MAC Address filtering. Some of the older models only supported WEP - Wired Equivalent Privacy which only provided rudimentary security, and indeed could be hacked by a determined hacker or enthusiast.
Should you choose a Wired or Wireless version of your Internet Connection Gateway? The choice is yours, but bear in mind that for a little extra money you get a model that has both capabilities. You may need to learn a bit more about how wireless networks operate and the additional security measures needed to fully protect such a network.
For home consumer Broadband Routers I would recommend Netgear or Linksys, and for business routers then you cannot go far wrong with Cisco.