Brake lights and tail lights/taillights are both red lights at the rear of most vehicles, but function at different times.
Brake lights are the red rear lights on a vehicle which either light up or get brighter when the brake pedal is pressed. More specifically, the brake light switch completes the electrical circuit that powers the brake light bulbs. Many of them also contain the rear turn signal lights as well. Working brake lights are mandatory on all vehicles in most regions, and some regions also govern the hight, location, and brightness of them.
Why are Tail Lights Red?
These lights are red because it's the color which most quickly draws human attention, and having it a different color than the headlights helps drivers know which direction the vehicle is facing at night. In addition, most traffic-related objects which are red mean "stop" (red lights at an intersection, stop signs), and braking is the process which brings the vehicle to a stop, so a psychological connection is made when brake lights are seen.
Because of the importance of the red in determining a vehicle's direction in poor visibility environments, having headlights in any shade of red is illegal in many regions.
Rear Lights Don't Have to be Red Anymore by Cars Simplified
Taillights don't have to be red anymore! Rear lights don't have to be red any more! The reason why rear lights look the way they do is because incandescent bulbs don't have the ability to emit a red light, so it has to be filtered in some way, and red coated bulbs never really caught on. Hopefully in the future we will see some style improvements which will incorporate less red in the taillights! Some vehicles have already taken this approach to some degree, like some Subaru Impreza models.
Tinted Brake Lights
Some aftermarket companies sell tinted taillights which appear to be dark red or completely black. These are designed and bought for aesthetic purposes, and aren't road legal in some regions. Some of them are less bright than their fully transparent counterparts, but are often fitted with upgraded lights to overcome the difference. Extra bright LED lights generating the appropriate color are often the only way to shine light brightly enough through a tinted lens.