The routers that the ISP's use do not log where people go. Too much processing power would be used to handle that task and it would have to stored off the router. Your typical router that an ISP or Enterprise would use looks at the destination address, compares that with it's routing table, then looks at the filters that are setup if any and then sends the packet on or drops it. It does this for every packet that comes in. You can set filters up that restrict access to certain addresses, address ranges or port numbers. HTTP is port 80, telnet is 23, etc. There are tools that can accomplish that task though.
As for home, yes, they typically provide more tools then an Enterprise or ISP router would have. There is no difference in an ISP or Enterprise router. So large companies have the same router a medium sized ISP would. Backbone providers have the largest routers that are made, as they have the bandwidth that could use the power they provide.
An ISP can use a Sniffer to capture packets. They can either capture packets going from Host A to Host B or from Host A to anybody, or anybody to anybody. These have storage limitations and speed issues. You can capture the whole packet, partial packets or just the header. The header will show the source and destination only. The whole packet is exactly what was sent. These are typically used for troubleshooting problems; either connectivity issues or to see who is a bandwidth hog. To use one of these on a LAN, it either needs to be plugged into a hub or a managed switch where you can set one port to be a span port. On the WAN side, you need physical access to the circuit, the proper cable and adapter. This allows traffic to pass through unaffected and it captures what is sent. With fast connections, it is not hard to fill a buffer up with LAN or WAN traffic. Capturing a specific machine is more ideal then everyone.
As for as home. Yes and no about seeing what sites they access. If you are talking about Internet sites, it depends. If they are going to a proxy, you will not see what site they are going to, the proxy knows though. If no proxy is used, and the router supports logging, the yes, as long as the buffer doesnt get filled, then older entries will get overwritten.
If you really wanted to monitor what people access, setup a proxy server and only let it access the Internet. Then you could use the logging features to see what they access.
How are you routing things through a browser? or did you mean router?