The cyclohexane chair conformation is one of the most stable structures in organic chemistry. The reason for this stability is that the carbon-carbon bonds are not strained by steric interactions. Instead, they are relaxed so that the electron clouds of the atoms involved in the bonding can occupy their optimal energy minima. In addition, there is no torsional strain in the chair conformation. As such, the cyclohexane chair is a relatively low energy conformation for this particular molecule.
However, it is not the only chair conformation for a given compound. Another conformation is the twist boat which also has good steric properties. For example, if the methyl group is placed in either the axial or equatorial position in the cyclohexane chair, it can be twisted around to give the twist boat conformation. The torsional strain is relieved by moving the flagpole H further away from the methyl group. This results in a smaller electron cloud for the molecule and a more compact cyclohexane structure.
Which chair conformation is more stable depends on the type of substituent and the steric effects. In general, the larger the substituent, the more it will prefer the equatorial position. This is because a large substituent will create a lot of 1,3 diaxial interactions. In order to minimize these interactions, it is best if the substituent is in the equatorial position.
This is especially true when the ring is heavily substituted. As the number of substituents increases, it becomes harder and harder to avoid a 1,3 diaxial interaction. As such, it is easier to predict that the conformation with the largest substituent in the equatorial position will be more stable.
The most significant difference between the cis and trans isomers of a cyclohexane molecule is that one chair conformation has both atoms in an axial position while the other has both atoms in an equatorial position. The cis isomer will be more stable because it will have no 1,3 diaxial interactions.
In contrast, the trans isomer will have two atoms in an axial position which will cause significant 1,3 diaxial interactions. Therefore, the cis isomer will be less stable. To change the cis to trans isomer of a cyclohexane requires breaking and reforming the double bond between two atoms. This is a much more complicated process than simply rotating single covalent bonds, which is why it is not as commonly used as cyclohexane rotation.