The Fasces was a bundle of twenty rods tied together, usually with an axe in the middle. They were considered divine symbols of a Roman magistrate's auctoritas, his authority. The twenty rods represented the original twenty Tribes of the Roman people. The rods also represented the symbolic authority of a magistrate to chasten physically. A Roman citizen could be beaten with rods, but not whipped. The axe was inserted into the fasces to symbolize certain magistrates' power to execute. In the days of the Roman Republic, the axe could only be inserted into the fasces for magistrates who left the pomerium (the sacred boundaries of Rome).
Not all magistrates had lictors (sacred attendants/bodyguards) and only lictors could carry the fasces. The use of fasces seems to go all the way back to the first Kings of Rome. Most of them were actually Etruscans, and it's thought that the tradition goes back to them.