First off, I'm assuming that, by computer chip, you are referring to a microprocessor or microcomputer on a single integrated circuit - although some people have used the term "computer chip" to refer to integrated circuits in general.
A microprocessor or microcomputer integrated circuit looks something like this:
The computer chip in the diagram above has several sections that are labelled things like cache, interface, FP control and Integer control, FP multiplier, and datapath. These kind of reflect the types of things involved in microprocessors: memories, interfaces, logic and calculations, and control. The caches, for example, are memories.
A computer chip works by allowing information to enter from the outside world (some of the interfaces), storing the information in the memories (like the caches), doing logical and mathematical operations on the information (like the FP or floating point multiplier), storing the results, doing more logical and mathematical operations on the information, until finally the information is outputted or causes something else to happen (some of the interfaces again). All of these activities need to be controlled (some of the parts labelled "control" may be involved), and this control is generally dependent on other information entered in the computer, called the programming (generally the software).
The information in a computer is stored and operated on (logical and mathematical operations) in digital form - that is, in the form of a set of 1's and 0's. Mathematics on 1's and 0's is referred to as binary arithmetic, and the logical operations are called digital logic.
Please click here to go to a website that has material from a course I developed for Motorola, which explains - starting at a pretty simple level - how a computer chip works. The course had the students use a shareware program for the Mac, called Digsim, to virtually "design" the computer chip on their computer step-by-step. If you have a Macintosh, Digsim can be downloaded from this site; if you have a PC, there are similar programs like MacroSim and WinBreadboard that you can choose and dowload at www.download.com
I think that, if you go through this website , you will understand how computer chips work. If you run into problems, or have questions along the way, feel free to email me at ericm@home.com. I do travel a fair amount and am rather busy, so please be patient if it takes me a week or more to respond.
You also asked: "What is the difference between chips of various companies? How do they vary in their working." Sometimes, there is very little difference between chips of various companies - companies often strive to make the chips look similar to the chips that their competitors make. These are often referred to as "pin compatible", meaning that the same inputs, outputs, power supplies, and other interfaces that are on one chip are in the exact same place and should work in about the same way as the other chip.
However, different companies generally have different semiconductor processes (how the computer chips are made), so there might be slight differences in the parts that have tried to be the same. If the companies agreed to make the parts similar (called "second sourcing"), then the designs used to make the parts will be nearly the same. However, if one company is trying to make a similar part without the agreement of the other company, then the company may have to use different design engineers to try to make their own version of the same part - and it might have a lot more differences.
Sometimes, different companies are not trying to make parts that are pin-compatible - they want the customers to choose their part over the parts from their competitors. By not making them pin-compatible, the company can try to be "sole source", and be the only supplier to their customers.