Which pest is this on my peppers?

It sounds like red spider mite, which are actually brown with two dark spots on their back. Only the over wintering females are red.
The bad news is that by spraying with water you just tend to knock them off the plant, leaving them free to colonise more of the surrounding plants.
If you only have a few you can try and squash them, but generally when people notice its quite a serious problem. I am assuming you want an organic solution as opposed to chemical so I would try a hot pepper wax which will smother the adults on the leaves but wont harm the eggs. You therefore need to repeat the process every 4 days or so untill all evidence is gone.

Hot pepper spray as follows:

½ - 1 ounce Garlic
2-3 ounces of onion
½ - 1 ounce of cloves
½ - 1 ounce of Cayenne pepper

Whiz in your blender with 1 Cup of water till very fine.

Add this mixture to one gallon of room temperature water. This spray must be repeated 3 times at 5-day intervals to kill off freshly hatched mites (the spray will not kill the eggs). If the mixture gets washed off it must be re applied.

For next year prevention is far easier than cure, google it and you will find the necessary steps needed to avoid future problems.

Good luck!

Thanks for the help.
I took the plant outside last night and attacked it with the hose on a very harsh spray setting, which seemed to remove most of them. I'll get to making the spray when I get in later.