と has a lot of uses. 

As far as I know, と can be uses to point to:

 - a member of a complete list (**X と Y と Z** => *noun X AND noun Y AND noun Z*)
 - a cause of a natural consequence (**condition A と natural consequence B** => *ALWAYS WHEN condition A THEN consequence B*)
 - a partner also doing the action (**person A と action Z** => *to do action Z TOGETHER WITH person A*)
 - with certain type of adverbs (the so called **"adverbs taking the 'to' particle"**), for example しっかり、だんだん etc.
 - a quote (**X と [person A は] said** => *"X" said [person A*])

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The と used here is a quotation particle:

`~と言う` => to say that ~

`~と約束する` => to promise that ~