I have a 1998 Nissan Sentra. When I break, only when I break, my left tail light turns off. I have changed the bulb to no avail. What is it ?

Get a test light and start checking power and ground at the left tail light with the brakes on.  Find out which one is missing.  Good luck.

A Nissan Sentra is an embarrassment . Get a decent car.

Stop breaking things.

I had a similar problem on an old minivan. It had 4 bulbs in the tail light assembly

1. Taillight

2. Brake light (separate brighter bulb)

3. Separate amber (yellow) turn signal bulb

4. Reverse bulb ( white)

If only one bulb was on, it worked properly, but if you used the turn signal while braking or hit the brakes with the tail lights on then things got weird. Any 2 or more bulbs were on at a time they all got dim. If one of them was the turn signal then all the bulbs would sort of flash from dim to a little brighter (but not all the way off and not to full brightness).

Anyway the mechanic figure out that the pigtail connector had gotten hot, the plastic casing had melted and caused a bad ground connection.

The connector had 5 wires: 4 "hot" wires for the 4 bulbs and one shared ground. When only one bulb was lit it forced the electricity through the ground connection even though there was some added resistance in the circuit. But when multiple bulbs were lit the electricity could flow through one bulb and back through another circuit (taking the path of least resistance) and that was why I'd get multiple bulbs  lit up or flashing with the turn signal.

Bottom line was that the mechanic replaced the connector and once there was a good ground connection with no resistance all the lights worked properly again.

So I would start by checking the ground wire. if there's resistance on the ground circuit it will make light bulbs do funny things.

You should learn to use "brakes."

Was the tail light bulb the same bulb for the brake light? If it's a double filament bulb you may have put the wrong bulb back in. 

You are getting a poor chassis connection on your rear light clusters.  Take off both clusters and clean up the corrosion on the metal parts.