There are a lot of different options you have when it comes to washing an egg.
The USDA recommends you use a cleaning solution to make sure you remove any bacteria from the outside of the egg. In feed stores, you can often purchase a solution designed specifically for washing eggs. Some of them allow you to wipe down the egg without removing the outside layer, the cuticle that keeps the egg protected from bacteria penetration.
However, you can just use water.
When I bring my eggs in, I look for any that might have spots or dirty places that need washed off. I just wipe them off with a paper towel under water, making sure I get everything nice and clean. If the egg looks clean coming out of the nesting box, I often won’t wash it at all.
Once washed, I’ll dry them off, put them in an egg carton and get them refrigerated right away until I’m ready to use them.