For metals, the compressive strength is the same as the tensile strength. However, there are normally two values given for tensile strength: the "yield stress" and the "ultimate stress" or "tensile failure stress" . The ultimate stress is the stress at which a tensile test specimen will break. The yield stress is the stress at which the piece will deform permanently. This is usually considered the failure point for metals, as permanent deformation usually makes a structure unsafe for further use.
To compute the failure load in either case, you multiply the stress by the cross-sectional area of the piece; i.e., a section perpendicular to the direction of load
However, a structural component under compression can fail in different ways than a tensile member. In the first place, it doesn't break, so the ultimate stress is meaningless. Once it yields, or deforms permanently, it is failed. But it can also fail by buckling at a stress less than the yield stress. Buckling is a sudden collapse caused by a deflection of the member from its original shape, and is a complicated and not easily predictable phenomenon. A cylinder can buckle in two different ways, depending on the proportions of its length, diameter and thickness. So if you are designing a long, slender, thin-walled cylinder you can't just use the simple formula above.
That said, you don't usually have to worry about buckling unless the cylinder is very long and thin. If the length is less than 50 diameters, and the diameter is less than 50 times the thickness, it should not be a concern, and you can use the yield stress to calculate the failure load.