What's the meaning of the phrase "excuser et s'excuser"? Why is a s' used before the verb?
(It is the title of a chapter in my French book, if the context is needed.)
S' is the contracted form of se, which is a reflexive pronoun. French uses reflexive pronouns with a fair number of verbs, more often than in English, where "verb oneself" is pretty rare. For example, French sometimes uses reflexive verbs to render explicit what is implied in English. This can be illustrated with the phrase "I'm going to wash", meaning that you are going to wash yourself, translated as "Je vais me laver", where me loosely means "myself". Moreover, French will pretty much always use reflexive verbs when English does (to perjure yourself/se parjurer), with the possible exception where expressions are used to translate verbs (to enjoy oneself/*passer un moment agréable). In other cases, French simply uses reflexive verbs where English doesn't (to remember/se souvenir; to apologize/s'excuser).
Excuser et s'excuser would be "to excuse and to apologize", but since it's a chapter title in a French book, it could as well be "pincer et se pincer" ("to pinch and to pinch yourself"), as it is likely the chapter on reflexive pronouns and will teach you how they work.
Reflexive pronouns are declined per person. They as as follow:
“S'excuser” is a pronominal verb. “Excuser” means “to excuse”, but is most often used as “s'excuser”, meaning “to apologize”.
You can see this question for more on pronominal verbs.