Red and Blue only effect the player that last hits it. They get a Red or Blue ring around them indicating they have it. These buffs are transferred on being killed.
Dragon buff stacks up to 5 times, with the first 4 being permanent. The fifth wears off, and if you are dead when your team gets it or die after they get it you lose it early. These buffs do not transfer on death.
Baron buff is similar to the fifth Dragon buff in that you don't get it if you were dead when Baron was killed and you lose it early if you die with it. It's not transferred on death.
It is possible to have someone with Red buff, Blue buff, Dragon buffs, and Baron buff all at the same time.
In general, your teams APC should take Blue buff, as it gives them cooldown reduction and mana regeneration. Your teams ADC is the best person to have Red buff later in the game.
There's no universally best time to take Dragon or Baron, but the general rule of thumb is to take them where there enemy team has people back at base or dead. This is especially true for Baron, with the ideal being multiple people dead.