I got the update for the steam client not long ago, and now when I double click on the Skyrim entry, it immediately pops up an error that the Skyrim Launcher stopped working. How do I make it work again?

Are you sure Steam was updated properly? Reinstalling Steam might be useful.

Delete the ini files that you'll find in My Documents/My Games/Skyrim. Verify the intergrity of game cache in Steam.