Objects (like comb) can be charged by rubbing as charged particles, particularly electrons, are transferred from one object to other. This can be seen as the object (comb) attracts small bits of paper. After some time, the charge on the body seems to disappear. How does this charge disappear without any external influence?
Charged objects have an electric field in their vicinity. The air will always contain a small number of ionized particles - this can be a result of cosmic radiation, local electrical activity, or just the chemistry of molecules. Now if your comb is positively charged, negatively charged ions will be attracted to it and positive ones will be repelled. Over time, this results in a neutralization of the charge on the comb.
The question then becomes - what is the major contributor to the conductivity of the air? A 1988 research report by Hugh R Carlon, Electrical properties of atmospheric moist air: a systematic, experimental study had the following in its abstract:
Reproducing figure 3 from that paper:
you can see that they measured a charge carrier density that is logarithmically dependent on relative humidity - and charge carrier density scales with conductivity (all other things being equal).
This is the reason, incidentally, that most ESD (electrostatic discharge) damage occurs in winter: when the air outside is cold, the relative humidity inside (where it is warm) will be low; thus the conductivity is low, and objects have a chance to build up charge without it leaking away so quickly.
One more figure from the same paper, showing the discharge current measured for different voltages, at different levels of humidity. Once again, it is hard to escape the conclusion that ionization of moisture in the air is responsible for the conductivity (and thus for the discharging of your statically charged comb):
Incidentally, in a dusty environment (see @Boris's answer) you may observe neutral dust particles being attracted to the comb (because of polarization of the particle) - but once it touches, it either sticks, or it will "jump away", having acquired a small amount of charge and therefore being now repelled. In so doing, these particles can be responsible for some charge leakage. But in clean air, moisture is the key component of conductivity.
Of course there is. You have the air around you, don't you? Nature does not particularly like a buildup of charge, and will try anything to get rid of it. In physics terms, a buildup of charge will have a high value of electric field nearby ($\rho=\nabla\cdot\bf E$ of course), and this large electric field will lead to the charge getting forced out.
Insulators aren't conducive to the motion of charge, they tend to get charges embedded in them. Yet, charge still can move--and thus leak off into the atmosphere. Once there, it will automatically spread out and subside.
The charge may also leak a bit to the interior of the insulator--but this won't change anything.