In the 19th century chess was conducted in an entirely different fashion that it is today. There
were few organizations and those that did exist were relatively weak and usually local rather
than national or international.  Since there were no organizations that could standardize all the
rules, conducts and conditions, each confrontation between players was generally unique.
 
Before the latter part of the 1800's there had been very few tournaments (for the same reason that there were no organizing bodies). The London Tournament of 1851 was the first international tournament and it was really a series of matches in which the winners of the first series of mini-matches would play one of other the winners of that series, etc., until one person was left standing.  The 1^st American Chess Congress followed a similar course. But soon some tournaments started experimenting with other approaches and with variables such as the use of timers.

Eventually the tournaments evolved. But in the time of Morphy, they weren't held in any particular esteem. Morphy himself felt they turned chess from an estimation of skill into a gamble of luck. They were enjoyable but not as conclusive as a match would prove to be.

So, it could be concluded that until perhaps as early as the 1870's but probably even later,
tournaments were somewhat negligible as compared to match play. There were too few tournaments for a player to make a living, or even expense money, from and they didn't enhance one's reputation as much as match play.

Match play was certainly the main conveyor of chess up until the 1870's and as such, there was a certain decorum about them. This is where stakes and odds enter the picture.

Stakes were money put up by both players, usually obtained from backers, and employed just like a bet - winner takes all. The purpose for the stakes was multi-pronged. First, they ensured the match was taken seriously and that the opponents were sufficiently credible that someone would back them. Second, some of the best players were professional and made a portion or all of their living through playing. Third, the stakes were a guarantee that the match conditions would be met. Fourth, the stakes helped defray the expenses, particularly in a prolonged match.

The custom of giving odds was an attempt to equalize different players' skill levels. In this way a lesser skilled player could reasonable play against a more highly skilled player. Stakes were usually required in odds matches also. Odds were also a measure of status, something like a rating and a way to determine whether one player was justified in challenging another player:  If player A beat player B even, and player C lost to Player B at odds, then player C had no justification to challenge player A.

 Occasionally the custom of odds giving was used, or rather misused, to preserve a players reputation. If a certain player refused to play even, he could never be beaten conclusively.
 

An interesting tidbit about chess played at stakes, which was a common coffeehouse activity in Victorian chess, often conjoined in the same arena of Odds, is that among some there was a queer house rule: If a player was mated by a pawn it was looked upon as so humiliating that he forfeited double the amount of the wager to the victor.

One of the chapters in on odds, and delves in greater depth than Soltis' book [Karl Marx Plays Chess -SBC]. Odds-giving, apparently, was the first informal rating system for players during the day. If a town or village included a chess master among its number, amateur challengers were given odds according to their respective ability. The old chess texts of the day assigned sobriquets such as "a rook-odds player" [or a Rook player - SBC] or "a pawn-and-move player" with the same implied denotation of strength as our "D player" or "A Player" titles today.

Odds was such a commonplace and accepted equalizer in Victorian chess that even openings books included whole swathes of opening theory to the "Knight Odds Opening," the "Pawn-and-Move Opening" and so forth. Louis Paulsen was even known to be studying openings for hours a day in preparation for undertaking a pawn-and-move match with Morphy (which never materialized). [Paulsen wasn't convinced that removing the queen knight's pawn was a disadvantage to the odds-giver -SBC]

The giving of odds has fashionably gone the way of the bowler hat and waistcoat, that is, rare and eccentric. Many players, even novices, feel insulted at being offered odds. There is an interesting double-standard at play, since most are adverse to receiving odds from the outset but absolutely relish the chance at snagging a decisive opening edge (which a would-be odds-giver can easily facilitate by playing garbage openings). Speaking for myself, I give and accept odds frequently when the gap in playing strength calls for it and my opponents are open to the possibility, though I'm the only chess player in my local cadre of players to do so.

According to Shibut's book [Paul Morphy and the Evolution of Chess Theory -SBC], odds-giving started to decline due to a couple of factors: The rise of the Soviet Chess Establishment in the 20th century which strongly disdained odds-giving in chess as garrulous and unscientific in its approach to the game, harboring back to its decadent image as a 19th-century pastime played at stakes in smoky gentlemen clubs and cafés  (and by the bourgeois nobles of the day). The second was the rise of the chess clock, which supplanted material odds-giving with time-odds, far more popular today. (personally I abhor time-odds, and find it doesn't matter as much, nor is it nearly as interesting, as giving material odds, especially when the odds-giver is rattling off nearly all his moves at breakneck speed anyway. That wouldn't be possible in odds-chess proper, if for no other reason that most openings need to be reappraised.)

There was a Morphy-Thompson knight-odds game where Morphy played a classically hypermodern Bird's opening, in such a style that strongly evoked Nimzovitch. It was especially fitting, actually, since in the Bird's opening the QN often has difficulty finding a happy home, and it's a closed system that deters early simplification. (and in that game Thompson was crushed)

Roughly 15% of Morphy's games were played at one form of odds or another. [My own research finds this to be closer to 25% -SBC] No other margin exists in any other major player's record [Staunton has the most recorded at odds games -sbc]. Morphy's mastery in taking the reins of incipient, disadvantageous positions in odds-games is another fascinating aspect of his historical legacy.

A pawn-and-move match against a player like Harrwitz or Löwenthal would have been especially interesting, as I believe those players employed a style that offered more resistance to Morphy than Anderssen.

 The excellent Karl Marx Plays Chess by Andrew Soltis is a collection of various chessic potpourri from his "Chess to Enjoy" column spanning the late 70s throughout the 80s. Among the various offbeat topics and anecdotes there is some info on odds-giving:

Pawn and Move: The odds-giver takes black and removes his f-pawn. Amusingly, some players thought this was in fact a sneaky way to give BLACK an advantage, via castling and a rapid attack along the semi-open f-file!

Pawn and Two Move: The odds-giver takes black, removes his f-pawn, and white moves twice before play commences normally. This form of odds requires a lot of finesse because 1 e4 & d4 give white an impressive command in the center and freer development lines against an already semi-nude king-side.

Knight: Odds-giver takes white and (usually) removes his QN. Sometimes not as much of a handicap and one might think, as there have been a number of brilliancies based on white's ability to swing his a-rook to the center faster with one less minor piece obstructing his way.

Exchange: A rare 19th-century specialty, and favorite of Blackburne. The odds-giver takes white and typically removes his QR and his opponent removes his QN.

Rook: Odds-giver takes White and (usually) removes his QR. Soltis pointed out an interesting stipulation that either the a-pawn or the h-pawn was also bumped one square so that the pawn is naturally defended, though I can't remember this surfacing in any of the odds games I've gone over. Apparently, castling is also legal; white just leaps his king to the normal castling square, whether or not a rook does likewise.

Queen: The toughest form of normal odds, though according to Soltis playing a rookless game requires more skill. A good queenless opening is: 1) d4, 2) Nc3, 3) Bg5, and 4) 0-0-0

Soltis also spoke about a Marshall Club experimental tourney in New York where the participants gave odds based on their ratings and the results were encouragingly not as skewed as one might think (the odds-givers only lost once to their lower-rated handicap takers). Material isn't everything, apparently.

Opening theory also was a different bad of tricks. In pawn-and-move 1) e4 e5?? Is now a horrific blunder due to the devastating 2) Qh5+! 1) ... c5!? is actually an offer of a second pawn, also due to the queen check. 1) ... g6, Nc6, and e6 were common black approaches to pawn-and-move in 19th-century chess. Morphy favorites were 1) ... d6 with the idea of a Philidor-type position, and the whimsical ...Nh6!? which doubtless baffled a few of his victims.

Move: Weaker player plays White.

Draw: The weaker player wins if the outcome is a draw under the usual rules.

Time: Weaker player receives more time on his clock.

2 Moves: Weaker player plays White and starts the game with 2 moves, which may not cross the 4th rank.

6 Moves: Weaker player plays White and starts the game with 6 moves, which may not cross the 4th rank

Pawn & Move: The stronger player takes Black and removes the pawn at f7. This was the most commonly given odds during the 19th century.

Pawn & 2 Moves: The stronger player takes Black and removes the pawn at f7. White makes 2 moves, neither of which may cross the 4th rank.

Knight: The stronger player takes White and removes his Knight at b1.

Rook: The stronger player takes White and removes his Rook at a1. His Pawn at a2 is moved to a3.

Rook & Move: The stronger player takes Black and removes his Rook at a8. His Pawn at a7 is moved to a6.

Rook & Pawn & Move: The stronger player takes Black and removes his Rook at A8 and his pawn at f7. His pawn at a7 is moved to a6.

2 Minor Pieces: The stronger player takes White and removes two of his Bishops or Knights, of his choice.

Rook & Knight: The stronger player takes White and removes his Rook at a1 and his Knight at g1. His Pawn at a7 is moved to a6.

2 Rooks: The stronger player takes White and removes both his Rooks

Capped Knight: The stronger player takes White. He must deliver mate with his Knight that started at b1 or lose. The loss of the Knight or any normal draw situation results in a loss for the stronger player.

Queen: The stronger player takes White and removes his Queen.

Capped Pawn: The stronger player takes White. He must deliver mate with his Pawn that started at f2 or lose. The loss or promotion of the Pawn or any normal draw situation results in a loss for the stronger player.

Queen-side: The stronger player takes White and removes the pieces starting at a1, b1, c1 and d1.