Web site for PCB routing or PCB tools
Is there a web site that can be fed with the circuit and it would route it and would give me the final bitmap of the PCB?


Querying Google for "PCB Design Service" yields lots of offers, but obviously, you don't just feed them with the circuit but also with (probably a lot of) money. PCB design cannot be fully automated and will be done by humans.


  I'll be also glad if you guys point me out to some free software for such PCB routing.


I can recommend KiCad. This software can also work together with online autorouters (which is what you might be looking for) such as http://www.freerouting.net/, but beware that this no "circuit in, PCB out" magic machine. Unless you have a really high number of signals, the effort usesd in correctly configuring the autorouter to produce an acceptable result is much higher than routing by hand. Also you still have to place the parts, which is sometimes the hardest part.