[paul.fr]

What is it that makes stinging nettles sting? And why do "doc" leaves give relief?

[that mad man]

Stinging nettles contain acid (formic?) and the "doc" leaf is alkaline so one helps neutralise the other.

Bee

[Bored chemist]

A significant part of the reason that dock leaves make the pain go away is that they are wet. The evaporating water in the leaf's juice cools the skin and helps relieve the pain.

[WylieE]

Stinging nettles contain histamine which is injected into the skin.  This is the same thing that we produce during an allergic reaction.  So this produces a localized allergic reaction increasing the number of mast cells. 
Stinging nettles also contain serotonin and acetylcholine.
Acetylcholine is a neurotransmitter and is what "activates" the nerves to make it feel like it is burning (I think I've posted on it here before, but my neurotransmitters aren't working so well and can't remember where).
Colleen

[Karen W.]

We use milkweed here to relieve the sting from a stinging nettle. Is Milkweed the same thing as what you call Doc leaf?? Does it have any other names..?

[chris]

Rubbing a painful area also invokes local pain control - hence you could rub a stinging nettle sting with almost anything (excepting perhaps another stinging nettle) and it would bring relief. The basis of this pain control is Melzack and Wall's 1968 "gate theory" of pain, whereby large diameter low-threshold mechanoreceptors (nerves signalling fine touch, stroking or tapping sensation) can suppress the flow of information from small calibre nociceptive (pain-bearing) nerve fibres. That's why you tend to automatically rub a body part when you injure it.

C