Most likely a faulty turn signal switch, or possibly the wiring to it.
replace switch assembly.
If you have the same bulb for brake and signal, brake light power needs to be interrupted by the signal lights, The brake circuit has to go through the signal light switch so the problem could be there.
Check fuse, check bulbs, check brake light switch, check wiring and then check signal switch, in that order.
Vehicle side, Brown, running lights Yellow, left signal/brake light Green, right signal/brake light Black, ground Trailer side, white is ground
the turn signal switch is probably bad
Assuming the signal lights still work, no. But a bad brake light switch or a blown brake light fuse could.
I may know the answer if it is a full size Blazer. There is a ground wire for the socket that may be connected to an area that has rusted and no longer is grounded. Reconnect to a solid piece of metal for the light to work.
Check the brake light fuse, the bulbs, the brake light switch and last, check the signal switch.
The 2001 Chevy Blazer brake lights may not work and the third bake stays on because the current light needs to be replaced. A replacement brake light can be purchased at an Auto Parts store like Pep Boys.
The head light switch is shorting out ,or a bad connection
Check and make sure you have the correct bulb. The turn signal/brake light is a dual element bulb and if you have a single element it does not work properly.
Your brake light is probably brake fluid low or emergency brake on. Rear brake lights are replaced by removing the lens and replacing the dual filament bulb.
Check the brake light fuse. If you have 12 volts at the brake light switch I would suspect the signal light switch is at fault.