You just opened a whole can of nerd, my friend.
Should be stated that someone’s colors can drastically change between moments in time depending on their mood and state of mind, it’s a pretty flexible (and sloppy) system.
Here’s an almost bare bones explanations of the colors:
Red is depicted as fire and chaos in-game because that’s what you’re going to see in most battles, but it’s explicitly clear it’s mainly the color of all things intense and passionate, whether it’s love or hate, pain or ecstasy. Also the usual color for hot heads and loose cannons. 
White is primarily order, community and perseverance. Justice and(/or) Law being upheld no matter what moral grey areas it might circumvent in the process. Very much representative of holy powers (The Light) and healing.
Green is the simplicity of the natural order. The law of the jungle and cycle of life and so on. It’s the color mainly used to represent beasts and other creatures that are true to their instincts or some other primal direction. Shares a lot of similarities with white when regarding community and healing.
Blue is intelligence. Creative thinking and adaptability, often detached from emotion in favor of progress. That, or willing to forgo tradition or “common sense” in favor of their own rationality. Of course it’s also the color for water and ice.
Black is power above all else, pure ego. Overcoming any obstacle, crossing every line, and taking whatever you please, no matter the sacrifices. Very much the color of necromancy and other dark arts. Often used for poison, alongside Green.
Most complex characters are prone to having several colors and color changes over time, so I’ll try to list their colors in order of importance though it might be pretty split for some of them.
Garrosh - Red
Gul’dan - Black
Varian- White/Red
Thrall - Green, White, Blue
Alexstraza - Red, White
Medivh- Blue, White
Malfurion- Green
Tyrande- Green
Zul’jin- Green/Red
Rexxar - Green
Lich King - Black
Arthas - White, Red, and later on Black
Kel’Thuzad - Black/Blue
Anub’arak - Black, perhaps green
ETC - Red
Artanis - White, Red
Tassadar - White, Blue
Raynor- White, a splash of blue
Tychus- Red
Dehaka- Green
Abathur- Blue/Green
Kerrigan- Black early on, Red later, Green
Zagara- Green
Blaze- Red
Imperius- Red/White
Tyrael- White
Johanna- White
Kharazim- White
Nazeebo- Black, Green
Leoric- Black, White
Butcher- Red/Black
Diablo - Red
Mephisto - Black, maybe Red
Azmodan - Red, Black
It’s usually pretty cut and dry.
Some ones I’m having difficulty placing though:
Brightwing - Red?
Stukov - Red and Green? White?