[sjastro]

Originally Posted by Eratosthenes

....so the "-1/12" result has units of energy? (ie eV or J)

Neither. The units are in hw where h is Planck's constant and w is the frequency of the oscillator.

The ground state energy, ie n=0, is not equal to zero (1/2hώ) and yet the summation in the Polchinski reference is from n=1 to infinity. Is the ground state energy omitted?

The mathematics behind both Quantum Field Theory and String Theory involves the use of mathematical operators acting on the energy state which brings about a particular change.
One such operator for the Hamiltonian H in equation 1.3.30 involves a general term aⁿₓ. This operator drops the bosonic string into the next lowest energy level.
If you start off with n=0, the operator will drop the string into an n=-1 energy level. But there cannot be an n=-1 level as the ground state exists for n=0. Hence you start from n=1 which also includes the ground state for the string. Note that n=0 is the vacuum state.

The quantum energy states are distinctly separated by an equal energy level equalling hώ, which is a small number, but nevertheless a finite number. So what happens when you sum an infinite number of finite numbers that increase by the same amount?

This has already been explained through renormalization.
The energy levels above a particular level are cut off as they exist above the energy threshold.
This is handled mathematically by multiplying each term in the sum 1.3.31 by an exp(-n) factor. As n becomes larger, the exp(-n) factor becomes smaller. For large n, exp(-n) is approximately zero.
The infinite series is truncated. Remaining terms can be further cancelled out by symmetry leaving 1.3.35 which equals 1.3.31.

The negative sign as well as the 1/12 result needs to be physically explained. What exactly does it represent?

It's been explicitly explained in the reference as.

This finite remainder is an example of a Casimir energy, coming from the fact that the string has a finite length.