Loathe is a verb meaning to dislike greatly. For example, if you have a mean boss, you might say that you loathe him.
Loath is an adjective meaning unwilling or reluctant. For example, you might say that you are loath to to spend time with your mean boss outside work.
Loth is a variant of loath and serves no purpose of its own. It is most common in U.K. English, though even U.K. writers prefer loath by a significant margin.
If you have trouble remembering the difference between loathe and loath, try to associate the verb loathe with breathe, clothe, bathe, and teethe. Each of these has a corresponding noun that lacks the e. Loathe is a little different because loath is an adjective, not a noun, but this works as a memory trick nonetheless.
I loathe Valentine’s Day. [The Trentonian]
But as much as I loathe technology, I need to know when writing letters will finally become obsolete. [The Maine Campus]
I had traditionally been loath to join a board game unless coerced. [The Brown Daily Herald]
Yet conservatives are loath to credit Harry Truman, who devised the original policy of containment. [BusinessWeek]