Brake/tail light when driving at night with the headlights on the right brake light would not work..thinking it must be the bulb I replaced & now the tail light wont work (right side) & brake light will only work when head lights are not on ...any suggestions???

Your year--make--model didn't post for some reason? That info would be helpful, to check wiring.
I'd probably check voltage and ground at the bulb socket.

Replace the right rear brake light bulb. It is possible that the filament has burnt out and has fallen across to rest on the brake light filament. This would cause the condition you are experiencing.

Check your fuses. I had the same problem with a 90 Honda civic. Dash lights went out, tail lights also went out. Replaced 1 fuse and all came back on.

Check for power with a test light(not a dvom) at the tail lamp assembly. Test right at the spot where the bulb socket sits. If you have power then i would check the connection tab on the bulb socket itself for pits or corroded tabs. If you do not have power present at test point then double check the fuse chart and see if it says 'electronic' instead of having a fuse amp rating for the tail lamps. If it says electronic for any of the light circuits then it means their is no external fuse that can be replaced and the problem is most likely internal of the headlight switch. Since the headlight switch is actually a power source for the lights. I would start there at this point. Let me know what you find and if you need anything else just let me know

there must be a short circuit which powers tail lights and brake lights. probably you have a dual filament bulbs which have both brake and tail light together. check for bulb, socket/or bulb housing and harness of that light.

Mr Pinske,

The rear running lights aren't on the same circuit as the headlight(s). To get the car legal for
night driving, replace the broken headlight (both need to work to drive legally at night),
and tackle the running lights as a separate project.
First thing to check on the running lights - the bulbs. Why: incandescent light bulbs that are
on at the time of an impact cannot withstand the impact and the filaments can fail, so fixing
your rear running lights may simply be a matter of replacing the two light bulbs.
If the bulbs are ok, your accident has caused a wiring fault, which you will locate
between the ignition switch and the rear running lights.
Here's the manual on how to replace the headlight:

1993 Buick Century
HEADLIGHT REMOVAL & INSTALLATION
Non-European Style
See Figures 1, 2, 3, 4 and 5

1. Remove the headlamp trim panel attaching screws.

Fig. 1: Remove the headlamp trim panel attaching screws

1. Remove the headlamp bulb retaining screws. Do not touch the two headlamp aiming screws, (at the top and side of the retaining ring), or the headlamp aim will have to be readjusted.

Fig. Fig. 2: Remove the headlamp trim panel assembly from it's mounting location

1. Using a screwdriver, remove the retaining ring screws.

Fig. 3: Using a screwdriver, remove the retaining ring screws

Fig. 4: Once the retaining ring screws are removed, the ring can be pulled away

1. Pull the bulb and ring forward then separate them. Unplug the electrical connector from the rear of the bulb.

Fig. Fig. 5: Pull the bulb out and unplug the electrical connector
To install:

1. Plug the new bulb into the electrical connector.
2. Install the bulb into the retaining ring and install the ring and bulb.
3. Install the trim panel.

Replacing the European style headlights is a slightly different procedure.

It may sound silly, but check your brake light bulbs to see if they have been installed incorrectly. Brake light bulbs have two filaments, one for tail light, and a brighter one for brake lights. On some cars if the bulb is installed wrong, the brighter filament will stay on for tail lights. The only other thing I can think of is the wrong bulbs have been installed.

The relay has nothing to do with the tail lights, but replacing the switch sounds like a good idea. There's a nut holding it in place, see http://www.skandix.de/en/spare-parts/electrics/switches/switch-headlight/1027524/ to get an idea of the construction. And the price. This is not worth fiddling about unreliable workarounds...
In addition, if fuses 15 and/or 16 burn frequently there might be a problem with the tail light cabling.

Good luck
Ingo

Try removing the bulbs from the rear tail lights you have one 2 bulb system shorting out, so one at a time remove the bulbs then replace the fuse and see which one is bad or just buy new bulbs and replace, also could be front running light blinker light. But sounds like brake light to ground threw bulb.

There should be at least 2 bulbs in the light assembly. One light is the light that is on when you turn on your headlights with out the brake pressed, the other light only lights up while you press on the brake. Make sure you are replacing the correct bulb. You may have been replacing the bulb that stays on. Your lights may be set up to turn off the light that stays on when the headlights are on while pressing the brakes.