Warm boots - are pretty much a reboot. You restart the computer after it has been running, or power off then power on. For example, if you installed a new driver for windows, and you restart the computer, this would be considered a warm boot. Also, called a soft boot.Cold boots - are first time powerups, powerups after a long time, or a power on after a shutdown, for example if you turn off your computer for the night, and power up after you wake up, this would be a cold boot. Another example is if you overclock your computer too much, and you have too turn off power via power button, and hold for a few seconds. Also, called a hard boot.
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