Hi Bomba,
Very basically, routers are devices that are used to connect two different networks together; allowing computers on one network to communicate with computers on other networks. Without routers, computers on any given network can normally only communicate with other computers on that network.
It's like dialing telephone numbers- without dialing an area code, you can only connect to other numbers in your local calling area. Think of that local calling area as your local network.
To communicate with people who live outside of your local network, you have to dial an area code (or country code, for that matter) to reach them. When the phone company's system sees that you've included an area code in the number you've dialed, it knows to send your call through a certain piece of equipment which will lead outside of your local network to the specific area where the number you are trying to reach is located. For computer communications, IP addresses are the equivalent of phone numbers, and routers are (one of) the devices used to connect "calls" to their proper destination.
You can find routers in use in many places: ISPs have huge routers in their networks to connect all of their customers to the Internet. Businesses have routers in their networks to connect all of the company computers together, and also to allow those computers to connect to the Internet or other external networks. Small offices or homes with multiple computers will also have small routers in those locations to allow all of their computers to share a single DSL or cable Internet connection. Such home/small office routers, manufactured by companies like Linksys, Netgear, and D-Link, are often called "Broadband Gateway Routers".
As far as reliability goes, routers (like other complex pieces of electronic gear) are prone to freezes, crashes, burps, chokes, whatever-you-want-to-call-them. These problems can definitely cause loss of network connectivity for computers connected to them, but there's no real answer to your question of how often this happens.
Also, because routers are basically small, specialized computers, they are prone to the effects of certain viruses, worms, and other malicious attacks. Bugs and loopholes exist in the software that runs in routers, and those bugs can be, and are, exploited by "hackers". A Denial of Service (DOS) attack is one example of a common type of exploit used to overload and shut down routers.