Pawn promotion

I am a newbie seeking an answer regarding pawn promotion. During a learning game, my white pawn reached the end of the board and I was allowed a pawn promotion. I already had my two rooks in play. However, I was allowed to replace (promote) my white pawn with a rook, thereby giving me THREE rooks in active play. Is this right?!?!?

yes

a pawn can promote to whatever it wants. This way, you can have like 8 queens from pawn promotion

So how does one do this using a traditional board game when there are no extra pieces with which to play? I was always taught that a pawn could only be promoted to one of your own pieces which had already been captured by the opponent!

No, you can promote a pawn into a Queen, Rook, Bishop, or Knight any time it reaches the promotion rank (the rule you describe @skrainka sounds like a rule for bughouse chess variant, but not standard chess, maybe who ever taught you got the rules mixed up accidentally?). 

This is one reason that more expensive chess sets usually come with an extra Queen for each side, it is in case one side promotes into a second Queen early in the game. When you don't have a spare Queen available, the generally accepted substitute is to take a captured Rook and flip it upside down. This indicates the piece is treated as a Queen. What if in some rare scenario OTB, you do not have a captured Rook, or spare Queen to use? Typically, you can substitute anything for that piece as long as both you and the opponent agree to treat that substitute as a Queen. In blitz tournaments, it is common for a side to promote into a Queen and use a spare Queen from someone else's set, or to just underpromote into what ever piece is within reach because time is of the essence. 

Of course, these are only OTB troubles. On chess.com, you can promote into as many Queens as you can get. In real life OTB games, this is true by rule, but usually one doesn't have that many spare Queens laying around. They can still promote into Queen after Queen, but typically substitute pieces will have to be used at some point. 

(p.s. Also keep in mind how rare it is to promote into multiple Queens in the first place, so this rule seldom comes into play. Usually the spare Queen, or upside down Rook is suffice for the duration of the chess game).