When you tap Highlight you will see a list of 20 colors that you can select from. Just tap the color you want and the cache will be highlighted with the color you selected. You can of course tap Cancel if you do not want to add a highlight. Your selected color will appear as a bar down the side on the Cache Details screen and the map icon will have a small bar in the same color.
Highlighted state of every cache will be synced across devices if those devices are logged into the same iCloud account. If they are not, syncing will not occur.
If it appears as if your highlighting is out of sync, go to the More... tab->Highlighting and choose "Manually Sync Highlights"
If a highlighted cache is deleted then that cache continues to be tagged in the cloud for highlighting. If you ever open that particular cache again it will be highlighted. You can, of course, delete a highlight at any time.
You can edit the label of the highlight so it means something to you rather than just having the label as the name of the color. You can delete highlights on all caches by color or even all of the highlights you have. How to do this is explained in the section on Highlighting in the More Tab
See below for an example of cache highlighted in Magenta