My 1966 mustang rear brake lights do not light up when the brake pedal is stepped on.  However, when I turn on the headlights the rear brake lights do come on.  Also when I select  the turn signal (left or right) the front turn signal lights work the rear do not.  Please help me out!  

I assume that you have checked or replaced the bulbs.  They have two filaments in them.  It is probably the turn signal switch in the steering column and this site has an excellent article on how to check and replace it.  Somewhere...  

There's a difference between brake lights and tail lights, which come on. Check your bulbs (cause its a lot easier than replacing the turn signial swith) Next will be a new turn isgnal switch. There are several places online to download a wiring diragram, I suggest you do so in case you need a reference to hook up the new turn siganl switch. good luck!  

Before you throw money at parts you may not need, be sure you have a decent multi-meter.  Take some checks at each of the wires coming from the front to see if the bulbs are getting voltage with the headlights on, the brakes depressed and the turn signals on.  Regarding the brake lights, you just very well many have a burned out switch.  Look on the brake pedal rod and you will see the switch.  Have someone help you, and manually depress that switch with the car running, and if the brake lights don't light up, you might suspicion that switch,.  If not, then go up river and check at several points to see where the break in the line is.  I wound up replacing my whole rear wiring harness.  Also, make sure your grounding points are clean.  Sometimes that causes the lights to not work.  

Ok thanks everyone for their help.  Light bulbs are changed and when you step on the brake pedal the lights turn on.  However, when I turn on the head lights the brake lights will not light up when the brake pedal is pressed. WEIRD!   Please help again...  

Now that you've said that, (which BTW is exactly opposite of what you said in your opening post) you would seem to definitely have bad grounds for the rear lamp sockets. The reason the brake lights work without taillights on is that there is a ground on the other side of the tail lamp filiments somewhere. This is a very common problem of that era of car. Take the bulbs out and make sure the lamp bases aren't corroded. Do the same for the lamp base to light housing connection and finally from the light housing studs to the car's tail light panel. That should solve your problem. (or you can buy new tail light stuff from several on-line parts house)  

OK problem corrected!

It was the lamp base to light housing connection.  I replaced them and now everything works perfectly.  Thanks to everyone for the help.  

I changed the lamp base housing and know everything works perfectly.  Thanks againQ