Free Radicals for aging
Free radicals are damaging because their unpaired electrons (or not fully filled valence shell) makes them highly reactive species. They are often considered together with highly oxidizing "reactive oxygen species" (ROS) such as peroxides. They are especially problematic for cell membranes and DNA. In the latter they can react with (oxidize) heterocyclic bases.

As suggested first by Gerschmann et al. (1954), since oxidative damage to cells and cell structures is seen to accumulate with age, the corollary of this could be that aging is a result of oxidative damage. The very fact that cells have antoxidant enzymes such as superoxide dismutase (SOD) suggests that suppression of these species is important in the cell.

Many (but not all) ROS originate from mitochondria during cell metabolism. Therefore, it has been suggested that by regulating metabolism (that is, reducing the metabolic rate) the "rate of living theory" (Harman 1981) the rate of oxidative damage can be reduced. One way to do this may be calorie restriction. 

This idea seems to fallen out of favour though. Another possibility is that the rate of mitochondrial ROS has nothing to do with metabolic rate per se; rather it might simply be a longevity determinant.

However, having said all this there is evidence that ROS are actually crucial in a number cellular pathways. It may be that it is the deregulation of pathways managing ROS that can contribute to aging rather than simply damage accumulation with age.

Gerschman, R., Gilbert, D. L., Nye, S. W., Dwyer, P., and Fenn, W. O. (1954). "Oxygen poisoning and x-irradiation: a mechanism in common." Science 119(3097):623-626. 

Harman, D. (1981). "The aging process." Proc Natl Acad Sci U S A 78(11):7124-7128.