This is mainly related to the stresses generated by the forces you create from pedaling. Any frame will flex around the bottom bracket a bit in response to pedaling loads. This flex can be felt, and many riders assume that it is consuming (wasting) pedaling effort. Actually, that's not the case, because the metals used in bicycle frames are very efficient springs, and the energy gets returned at the end of the power stroke, so little or nothing is actually lost. While there is no actual loss of efficiency from a "flexy" frame, most cyclists find the sensation unpleasant, and prefer a frame that is fairly stiff in the drive-train area. This is more of a concern for larger, heavier riders, and for those who make a habit of standing up to pedal. Another area where lateral stiffness can be an issue particularly to the touring cyclist is the rear triangle, when there's a touring load on the rear rack. An frame that is too flexy in this area will feel "whippy" and may be prone to dangerous oscillations at high speeds. Most of this flex is usually in the luggage rack itself, but there can be enough flex in the seat stays to aggravate this condition.

(Since this article deals with frames, the issue at hand is road shock transmitted from the rear tire to the saddle. Ride qualities experienced at the handlebars are to some extent determined by the fork, as well as geometry, and flex in other bolt-on parts, but are un-related to the choice of frame material.) Much of the commonplace B.S. that is talked about different frame materials relates to imagined differences in vertical stiffness. It will be said that one frame has a comfy ride and absorbs road shocks, while another is alleged to be harsh and make you feel every crack in the pavement. Virtually all of these "differences" are either the imaginary result of the placebo effect, or are caused by something other than the frame material choice.
Bumps are transmitted from the rear tire patch, through the tire, the wheel, the seatstays, the seatpost, the saddle frame, and the saddle top. All these parts deflect to a greater or lesser extent when you hit a bump, but not to an equal extent.
The greatest degree of flex is in the tire, probably the second greatest is the saddle itself. If you have a lot of seatpost sticking out of a small frame, there's noticeable flex in the seatpost. The shock absorbent qualities of good quality wheels are negligible...and now we get to the seat stays. The seat stays (the only part of this system that is actually part of the frame) are loaded in pure, in-line compression. In this direction, they are so stiff, even the lightest and thinnest ones, that they can contribute nothing worth mentioning to shock absorbency.
The only place that frame flex can be reasonably supposed to contribute anything at all to "suspension" is that, if you have a long exposed seatpost that doesn't run too deep into the seat tube, the bottom end of the seatpost may cause the top of the seat tube to bow very slightly. Even this compliance is only a fraction of the flex of the exposed length of the seatpost.
The frame feature that does have some effect on road shock at the rump is the design of the rear triangle. This is one of the reasons that touring bikes tend to have long chainstays--it puts the rider forward of the rear wheel. Short chainstays give a harsh ride for the same reason that you bounce more in the back of a bus than in the middle...if you're right on top of the wheel, all of the jolt goes straight up.