Since the wu's here don't seem to have much on the actual internals of the routers I think I'll have a bash...
Basically the router is just a computer, much like the one that you are sitting at right now. The only difference really is that the router has one dedicated purpose; to get packets from A to B. I'll try not to be too model-specific here, I've only used the 2500 and 2600 series of routers so if anyone has any other experiance give me a shout.
On the lowest level the router consists of some components (a CPU, memory and interfaces), however it's not really that simple...
The CPU used is embedded, usually at speeds between 50-400MHz, depending on the task. Obviously a router controlling traffic between two small networks is going to need a lot slower processor than that of an inter-domain router on the Internet acting as a firewall.
The router infact has four different types of memory, these are Flash, ROM, RAM and NVRAM and each as a specific purpose.
The Flash contains the IOS, the operating system of the router, it can contain images multiple versions of the IOS at the same time and may be upgraded if needs be.
The ROM contains the bootloader and a simple, stripped-down version of the IOS. This is called the RXBOOT mode.
RAM contains any services that are currently running on the router (ie the http server etc), the running-config. It also has a copy of the ARP and routing tables as well as the buffers.
The NVRAM (Non-Volatile RAM) stores configuration files for the router, the most common use of this is to store the startup-config which tells the router what to do when it starts up.
A router can have a wide variety of interfaces on it. All routers (at least all the ones I've seen), have a console port.
The console port is an RJ-45 socket with a wire mapping for a rollover cable, this is used to configure the router from a PC using a DB-9 adapter to a serial port. This is much like having the PC as a dumb terminal and infact works the same way (you can use TeraTerm or HyperTerminal to log into the router).
Also there is an auxiliary port, this is attatched to a modem (also via rollover cable) so remote configuration can be done.
Moving on to the actual network interfaces used for the routing of traffic. The most common types of interface are serial, Ethernet and ISDN.
The serial port may operate in both DCE and DTE modes, allowing you to both provide as well as recieve at about 1.5Mbit/sec over a serial link.
Ethernet is probably the most common networking technology used today and rightly is included on most routers as standard. Most often it is the 100MBit/sec IEEE 802.3 Ethernet standard.
ISDN capabilities are provided by a BRI interface on lower models (Basic Rate ISDN) and PRI (Primary Rate ISDN aka, T1) on higher models.
Other interfaces on higher models include ATM and SONET along with other high speed connectivity devices. Token ring is popular as well as Frame Relay.
At this point it's worth mentioning the modular nature of the higher end routers. From the 2600's the routers can be supplemented with extra interfaces, much like a PCI expansion cards for PCs. Not only can new interfaces can be added but additional ones can be added to a router.
As I have said I only really have any experiance with the 2500 and 2600 models but cisco.com has a wealth of information on all of the router's different features if you are interested.