This is the favoured method of quoting odds by UK Bookmakers, fractional odds quote the net total that will be paid out to the bettor, should he win, relative to his stake.
Odds of 6/1 ( "Six-to-one" or "Six-to-one against") implys that the bettor stands to make a £60 profit on a £10 stake. The bettors stake is also returned making a total return of £70.
If the odds are 1/6 (read "sixr-to-one on"), the bettor will make £10 on a £60 stake. Odds of 1/1 are normally called evens.

Some bookmakers that offer odds in fractional format

Decimal odds are favoured in in continential Europe and Canada, decimal odds differ from fractional odds in that they quote the total amount that will be paid out to the bettor including thier initial stake.
So to put it simply, decimal odds of an outcome are the same as the decimal value of the fractional odds, plus one.
The 6/1 fractional odds used above would be quoted as 7 (1+(6/1)), while the 1/6 odds would be quoted as 1.167 (1+(1/6)).
Decimal odds are considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome bet.

Decimal odds are favoured by betting exchanges as they are the easiest to work with for trading.

Some bookmakers that offer odds in european format

Moneyline odds are the odds favoured by American bookies. If the figure quoted is negative, then the moneyline odds are quoting how much money must be wagered to win £100 - this is done if the odds are better than even. If the figure quoted is positive, the odds are quoting how much money will be won on a £100 wager - this is done if the odds are worse than even. Even odds can either be quoted as $100 or -$100.
Fractional odds of 6/1 would be quoted as (+)£600, while fractional odds of 1/6 would be quoted as -$600.

Here are an example of how odds are presented in different formats:

Some bookmakers that offer odds in american format