No, different brand routers dont't recognize each other.
to split the Internet between different computers or to make it wireless if it is a wireless router
There are many companies that make different types of wireless mimo routers. The router that has the most reviews is the Belkin model router. This would be the best router to use, and they make a different model based on price, and speed that you are looking for.
You don't "make" your router have wifi. It either has it or it doesn't. Wifi is available from a wireless router, so check to see if yours is wireless. A wireless router will usually, but not always, have antennae sticking out of it.
You can't add wireless router to your computer. You can a wireless router to your network. If you do so, it does make your network wireless. In order to use your network wirelessly you need to have wireless adapter for all your computers you want to be connected wireless.
You need to log in your router, and enable security for wireless networks.
The Belkin wireless router is a good brand. There is nothing specifically different or special about it but a lot of people prefer you use this brand because they make many good wireless products.
Many companies make a wireless router and the cost depends on what you are looking for. Try the Linksys or D-link wireless router for around $60.
to make it yours
Wireless internet is just done using a wireless router. You can use any type of internet and hook up a wireless router to make it wireless for your devices in your home.
Get a wireless router (N type is recommended). If you do not have wireless adapters you need get it. Then just read manufacture manual for the router.
Several companies make wireless TV routers. I personally use a cisco router, and it is very good quality.
You will need a wireless router and a wireless adapter to have wireless pc to pc connection