Router is a network device which provides ip routing according to its routing table. Routing table is the table, where the best routes to the destination networks appear. Router checks the source and destination addresses of the packet and sends it to receiver via dynamic ip routing protocols or static routes. Static route is the route that you configure manually. Dynamic routing protocols have algorithms and they route the packets due to that algorithms. Some of the dynamic protols are RIP , OSPF and EIGRP.
Basicly routers are used for seperating different Local Area Networks. They work on Layer-3 of the OSI Model. There are several types of routers such as wireless , xDSL , broadband or etc. Wi-Fi routers are used for wireless connection. xDSL routers provide DSL connection, they termine the p2p connections and routes the packets through a DSLAM or active ethernet switch into internet network. DHCP role is actively used on that type of routers (modem) to attach ip addresses, subnet masks or default gateways as well.
Mostly a router is connected to a switch on local area networks. And clients are connected to that switch. Router’s main job is providing inter VLAN routing or seperating different non virtual local area networks. If you are using a high level router, they are named as “core router” which other routers are termined on it via MPLS or SDH.
If you are preparing for CCNA or CCNP certifications you can simulate the routers using GNS3 or Packet Tracer programs. You can set up your own network and configure the routers.
Some of the well known router brands are Cisco, Huawei, Alcatel, TP-Link, Alcatel, HP, Inca and Motorola.