The key to understanding trends in ring strain is that the atoms in a ring do not necessarily lie flat in a plane. We begin by studying the most stable conformation of cyclohexane, which has completely staggered dihedral angles at each of the six C-C bonds. This conformation is not flat but is folded into the shape of a lawn chair, so it is called the chair conformation.
While the chair is typically drawn from such a perspective view, keep in mind that the chair actually has three-fold rotational symmetry. That is, it can be rotated by 120 or 240 degrees and look identical. The best way to visualize the chair conformation is to build one for yourself using a molecular model kit. You should verify that the chair has no eclipsing interactions. This can also be seen in a Newman projection down a set of two C-C bonds. Notice how this resembles two Newman projections of ethane joined together by -CH2- groups:
There are two distinctive types of C-H bonds on the chair. One set is comprised of C-H bonds that extend vertically up and down and are called axial bonds. The other set consists of C-H bonds that extend out to the periphery of the ring and are called equatorial bonds. Each carbon has one axial bond and one equatorial bond.
Recall from our discussion of cis-trans isomerism that each carbon has a bond that points up to the top face of the ring and a bond that points down to the bottom face. Don't get these confused with the axial/equatorial classification. One equatorial bond isn't necessarily cis to another equatorial bond, and the same applies for axial bonds. The up/down orientation of an axial position changes from one carbon to its neighbors. If axial is up on one carbon, axial will be down on its two neighbors. The same is true for equatorial positions.
The chair conformation is used so frequently that you should become comfortable with drawing them. The most important feature of chair conformations is that they consist of three sets of parallel C-C bonds. Begin by drawing the six C-C bonds that comprise the skeleton of the chair: Drawing the equatorial bonds in the correct orientations is probably the trickiest part of the process. A useful rule to remember is that each equatorial bond is parallel to one set of C-C bonds you have already drawn. The parallel C-C bonds are the ones that the current carbon is not attached to: Finally, fill in the axial bonds, which are just lines going up and down.
Like other conformations we have studied, chair conformations are in a state of constant flux. Because all the C-C bonds are interconnected, they cannot rotate independently but have to move together. For example, one end of the chair could "flip up" to put the cyclohexane ring in a boat conformation.