It's possible that your battery has enough charge to light the headlights and dash, but not enough to turn the starter. Typically though in this case you will hear some sort of click when you turn the ignition to "start." Your description says you hear nothing, however. That points me to consider that some sort of an electrical interlock (safety system) is preventing the car from starting.
If an automatic transmission, check that the transmission is fully in park or fully in neutral. It may help to shift through the gears and return to park. Make sure your foot is firmly on the brake when you turn the key. Automatic transmission cars won't start in anything other than park or neutral, and sometimes you need to depress the brake.
If a manual transmission, make sure your foot is fully on the clutch. Modern manual transmission cars require the clutch to be depressed before starting, to prevent the car from lurching forward when you turn the key.
In my mind, these are the simple solutions. If it's not one of these, the troubleshooting gets more involved. Though if you get this far and it still won't start, it doesn't hurt to get out the jumper cables to positively rule out a battery problem.
EDIT: Could also be a bad starter solenoid, or bad connection to the starter solenoid. See an earlier question and answer: Car Suddenly Decides To Not Crank