Bobby Fischer my 60 Memorable games.

In the 3rd game, Petrosian vs Fischer, Portoroz 1958 (King's Indian Defense)

Fischer mentions at the end that he offered the draw, not realizing it was bad etiquette.  It was Petrosian's place to extend the offer. 

I am relatively new to the rules so can someone explain this to me.  Is it because white should be the one who asks for draw or is it something else ?  I thought either player could request a draw.

Bobby hadn't proven himself at that level yet (it was just 1958, Bobby wasn't a monster until the mid 60s early 70s).  Petrosian was a former world champion.  For players like you and me it comes up in OTB tournament play (online it doesn't matter, anyone can offer a draw).  If you're lower rated, it's bad etiquette to offer a draw.  Because the stronger player "should" win, and knows more ways to test the position to see if you really can draw it, you should wait for them to offer first.  When they're out of ways to try for a win, and after you've proven it's a draw, they'll offer it.

When the weaker player offers a draw first it's kind of like saying, "I'm as good as you and I know it."  Not a big deal, and I can't imagine it would earn you any enemies, but it is a bit rude.  If you decline the draw from the stronger player, he wont offer again until you offer.  Because you've made it clear you won't accept a draw, etiquette says the next one to offer a draw will be you and so on.

I had always assumed that it was the player with the chances who ought to offer the draw.

That's how I interpreted it.  At least here in the United States that's how it is (or at least that's what I was taught...)  I assumed what I was taught is applied elsewhere, and didn't bother looking it up online.

In Ireland does it matter if the lower rated player offers a draw first?  Just curious.

That makes sense for nearly equally rated players.  I was told the lower rated guy shouldn't offer a draw first though... maybe it's an outdated thing heh, seems like I'm the only one who's heard of it.

Bobby just meant that he offered the draw before making his move.  According to the Laws of Chess (article 9.1 b), you're supposed to make your move before offering the draw.  After all, you could blunder.

Wafflemaster, you made a few mistakes. In 1958 Petrosian was not a former world champion. he won the championship in 1963. Bobby was pretty much considered a monster while still in his teens---in this tournament he became a GM and the youngest ever to qualify for the candidates tournament.

Your explanation of draw etiquette is confusing.

In this case, I don't think it has anything to do with the ratings of the two players (FIDE started publishing Elo lists in 1970) nor whether Black had played his move first or not.  In the final position on the board, Petrosian has a pair of united passed pawns, and it looks as if he might be able to queen one of them, so all Fischer is saying is that perhaps he should have demonstrated his drawing rook sac, and shown that he could queen his own pawn.  It is certainly not a major breach of etiquette, given that the drawing line was probably obvious to Petrosian as well.

Fischer also may be alluding to the fact that Petrosian had been a member of the world's elite for a few years by then whereas this was Fischer's first foray into the world championship cycle, so he should show respect for his opponent.  Most of the players at Portoroz were quite amused at how young Fischer was, so I somehow doubt that anyone took offence if he committed any minor faux pas's.

It is bad etiquette to offer a draw as soon as you realize you have lost, especially if it seems likely that your opponent knew that you were lost even before that.  Repeated draw offers are also considered rude.   You can offer a draw once in a position which you think is equal, and if they turn you down, you should either wait for them to make a draw offer, or for some major change in the position that makes the draw obvious.