1. His ambition to become the world champion was understandable.

(Waht is the part of speech of 'to become the world champion'? Is it a noun phrase or an adjective phrase?)

2. He has only one aim, to make a lot of money.

3. He has only one aim to make a lot of money.

(Do we have to put a comma before to.... or not? What is the part of speech of 'to make a lot of money'? Is it a noun phrase of an adjective phrase? I appreciate your help. Have a great weekend!)

1. I'd call "to become the world champion" an infinitive phrase, being used as an adjective to modify "ambition."

http://grammar.ccc.commnet.edu/grammar/phrases.htm#infinitive

2. I'd put a colon (:) after "aim" -- "to make a lot of money" is also an infinitive phrase, this time serving as a noun. I guess you could call this an appositive since it re-names "aim."

3. Incorrect -- needs punctuation after "aim."