First, move any 20 pennies to the other table, keeping the same side on each penny up.
Second, flip all 20 pennies on the other table.
Explanation
Initially table 1 will have 80 heads and 20 tails, and table 2 will have 0 heads and 0 tails.
Let t be the number tails moved, and h be the number of heads.
After the move table 1 will have 80-h heads and 20-t tails, and table 2 will have h heads and t tails.
After the flip table 1 will have 80-h heads and 20-t tails, and table 2 will have t heads and h tails.
So, at this point table 1 has 20-t tails and table 2 has h tails. We moved 20 coins: h heads, and t tails, so h+t=20, or h=20-t. 20-t and h must be the same amount, thus both tables have the same number of tails, although we don't know what that number is.
Thanks to Dan Florentin for this problem.
Michael Shackleford, A.S.A.