Hi... I am looking at install a kitchen sink and am wondering how far can my sinks drain be from the existing waste water plumbing.Does it have to be a specific distance or is there certain distances you can work with in?Will the S-trap allow me to adjust any distance in between?

To answer your question, there is really no limit to the distance your sink drains can be from your existing plumbing; however, you do need to take into consideration that you must have a 1/4" fall per/foot. Also, the S-Trap is not approved and considered the best method. I would recommend using a P-Trap, it follows code, and will be a better suit to help with preventing sewer gas from exiting. You may need to add an AAV (Studor Mini-Vent) to help with the flow of water exiting your sinks. Hope this helps.

Your "S" trap is a code violation so your question is mute.
To stay in code you will have to install a "P" trap and vent it. This can be done in several ways.
1) Run a vent up through the roof or revent back to a existing vent in the attic. In which case the sink would have to be within 5 feet of the vent with a 2" drain or 3 1/2 feet with a 1 1/2" trap and drain line. **OR**
2) Install a AAV,(Air Admittance Vent)(see image) just downstream from the trap Good luck, Tom