What does the damage shorthand in Nethack mean?

Solution 1:

This is standard D&D terminology, where the first number (x) indicates the number of times (or the number of dice) you roll and the dx indicates the type of die that is rolled (with the x indicating the number of sides). In the case of 8d8, you would roll an 8 sided die 8 times and then sum the results.

Therefore, the damage done by Touch of Death is the sum of 8 random numbers between 1-8, which works out to a number between 8 and 64. As per liori's comment, this value is more likely to be in the middle of that range than at the ends. If you take all the possible outcomes of 8 rolls of 8 sided dice and graph the sums, you'll get a very nice bell curve. Rolling 8 or 64 would be extremely unlikely, but rolling 32 would be extremely common.

A few examples:
1d20 would be 1 roll of a 20 sided die.
3d6 would be 3 rolls of a 6 sided die.
1d8 + 2 would be 1 roll of an 8 sided die, with 2 added to the result (so a number between 3 and 10).