What do Numbers represent?
A number is an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. It is also a quantity or amount; such as several, in a group, company or order (such as a magazine issue to indicate a position in a series).
In the LOST mythology, the Numbers were the glue that bound many clues.
We really first learn of the power of the Numbers by Hurley hearing them while he was at the mental institution. The Numbers were supposed to have been heard by a patient, who said they were cursed. But despite the warning, Hurley used the Numbers on a lottery ticket. So, the Numbers were at first, lucky. But as Hurley started to embrace his new wealth and fame, the winning lottery ticket became his own curse (with family members being hurt, people dying, etc.)
We also found out that the Numbers were broadcast possibly as an island location beacon to the DHARMA group. Why the numbers were important to DHARMA has led to speculation that the main purpose of island research was to re-set the Valenzetti Equation, a large doomsday-predicting formula on the demise of mankind.
We also found the Numbers stamped on the Hatch cover. This apparent serial number freaks Hurley out as a bad omen. In some respects, that was true. The Hatch discovery led to Desmond and the internal workings of DHARMA, and more mysteries and clues (such as the blast door map). Desmond was a lost soul also imprisoned on the island to do unexplained work for an alleged higher purpose. (One new theory is that the Hatch and electromagnetic fields were being operated by human souls in order to regulate the gateways between life, death, heaven and hell. The operators were not told of their role, least they could interfere and destroy the natural world.)
Finally, the Numbers are the code in which needed to be placed into a computer control every 108 minutes or bad things would happen (an electromagnetic build up would create a lockdown, a purple flash, release of energy, to dangerous explosion-implosion events). Why human beings had to enter the code to regulate the release of an alleged energy build up is unclear, but may take homage to the soldiers who man defense missile silos - - - who have to manually enter launch codes in order to fire destructive nuclear warheads. The idea is that these men and women have the final say on their own fate; and as a check against a computer malfunction.
So the Numbers represented good luck, bad luck, a key, a curse, and tie that bound many different elements of the story together. But in the end, TPTB merely said that the Numbers were a red herring, with no real significance to the main story. That reveal was one of the major disappointments to serious fans.