もらる is used when the speaker is the subject, and indicates that the speaker is receiving a benefit
あげる is used when the speaker is the subject, and indicates that the speaker is providing a benefit
くれる indicates that the subject is providing a benefit to the other party

For example:

	Me	Tanaka-san
(私は)田中さんに本を貸してもらう。 I had Tanaka-san lend me a book.	borrower	lender	私 is the subject of もらう: I received a favor from Tanaka-san. (Note: as is typical in Japanese, 私 would not be specified as it is the implied subject)
田中さんが私に本を貸してくれた。 Tanaka-san lent me a book.	borrower	lender	田中さん is the subject of くれる: Tanaka-san did a favor for me.
田中さん、お金を貸してもらえる? Tanaka-san, will you lend me (the) book?	borrower	lender	In this example, the Japanese literally means 'Tanaka-san, can I get you to lend me a book?'. In Japanese it is common to use かす where in English we would typically use かりる: 'Tanaka-san, may I borrow your book?'
田中さん、お手洗いを借りていいですか。 Tanaka-san, may I use the bathroom?	borrower	lender / user	In Japanese, when asking politely for permission to use the bathroom, it is common to use this form, which literally means 'may I borrow the bathroom?'
田中さん、トイレを貸してください。	borrower	lender / user	In the same way, it is common to ask for something using 貸してください - please lend me. This usage implies a more informal relationship than 借りてもいいですか.
(私は)田中さんに本をあげる。 I gave Tanaka-san a book.	provider	recipient	私 is the subject of あげる
(私は)田中さんに本をもらう。 I received a book from Tanaka-san.	recipient	provider	私 is the subject of もらう

Technically it is possible to combine 借りる with もらう, あげる, くれる etc in the same way, but these usages can imply somewhat convoluted meanings, so are not commonly encountered. For example, 借りてもらう literally means 'have someone borrow something (on your behalf)'.