Which is more stable, cis -1-ethyl-2-methylcyclohexane or trans -1-ethyl-2-methylcyclohexane?

The more stable of the two will be trans-1-ethyl-2-methylcyclohexane, and here's why that would be.

I'll compare the two most stable chair conformations for each molecule. If you want to see why these chairs are the most stable, see

http://socratic.org/questions/how-can-i-draw-the-more-stable-chair-conformer-of-trans-1-ethyl-2-methylcyclohex#119572

http://socratic.org/questions/how-can-i-draw-the-more-stable-chair-conformer-of-cis-1-ethyl-2-methylcyclohexan#119617

Now, here's the most stable chair conformer for trans-1-ethyl-2-methylcyclohexane

Notice that you have the ethyl group attached to carbon (1) in UP position on an equatorial bond. The methyl group is attached to carbon (2) in DOWN position on an equatorial bond as well.

Now look at the most stable chair for cis-1-ethyl-2-methylcyclohexane

Once again, the ethyl group is in UP position on carbon (1), but this time the methyl group is in UP position on an axial bond. The fact that the methyl group is on an axial bond will ultimately determine which of these two chair conformers are more stable.

Ideally, more stable chairs wil have the larger groups on equatorial bonds. As you can see, that is what happens in the first chair. For the second chair, the methyl group's position on an axial bond will cause steric strain, which will reduce the stability of the chair.

As a result, trans-1-ethyl-2-methylcyclohexane has a more stable chair conformer than cis-1-ethyl-2-methylcyclohexane.