The differences usually reside in actuation force (how much force is needed for the switch to register), noise level, and whether or not there is tactile feedback (a bump when you hit the key).
Cherry MX Reds are considered the "gaming switch" because they have the lowest actuation force, are basically silent, and have no tactile feedback. This is great for fast paced games that require very quick feedback.
Cherry MX Blues are considered the "typing switch" because they have high actuation force, are quite loud, and have sharp tactile feedback. They are very clicky and because of its actuation force it prevents mistyping if you accidentally fat finger a different key. The tactile feedback makes sure you hit the key that you want to.
Cherry MX Browns are the in between. They have a middle ground actuation force (45g in comparison to MX Blue's 55g and MX Red's 35g), they are silent like reds, but they also have tactile feedback.
All of the other switches, such as MX Clear or MX Black, are all versions that are based on the other three. For instance, MX Blacks are basically MX Blues with more actuation force.
This is a vast exaggeration between the switches, I should say. Most people prefer browns or reds, since most people find blues too loud or too exhausting to use for extended periods. It's mostly personal preference between them, and until you feel the switches for yourself, you'll never really know which one you like until you try them all. I have a K65 with red switches and I love it, although I had a Razer keyboard with blue switches and liked that one as well.