Well, let's see:
Nothing happens. I’m still alive and posting on eee-af.
Why? Lets measure the current. (My right finger is on the negative meter probe, and the meter is set to the 2 mA range)
0.026 mA; that's 26 micro amps, that’s 12V × 26µA = 312 micro watts of power being dissipated in my body (as heat).
Nothing is going to get burned with 312 micro watts.
You may wonder why a battery capable of providing 1000 amps, only provides 26 micro amps. Because of resistance, as other have said. Skin resistance is very high, and current is voltage divided by resistance.
Now, if I drop a wrench (with essentially 0 ohms of resistance, Yes I know it’s not really 0 ohms, but close enough for our purposes) across the battery, the current would be 12(volts) divided by 0(ohms), or infinite current. Of course, that’s not possible, but the battery will provide everything it can. This will probably be about 1000 amps. So 12(volts) times 1000(amps) is 12,000 watts. That’s a lot of power, and will easily melt parts of the wrench. A metal watch band or ring can get hot enough to melt, and seriously burn you if the battery somehow contacts it in two places.
So, your answer is nothing will happen because your skin resistance is way too high to conduct any significant current at 12 volts.
For the adventurous out there put a 9 volt battery on your finger and you will feel nothing. Put the same battery on your tongue and it will hurt A LOT. (Don’t say I didn’t warn you). That is because a wet tongue has much lower resistance than skin.
I am adding a picture of what can happen to a wrench.
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