What does the cross mean on the Christian flag?

The cross means that Jesus died for the sins of the world on that cross.

Another Answer:

For many 'Christians' the symbol of the cross 't' is the 'all in all.' The cross is looked upon as a grand charm, the great refuge in every season of danger, in every hour of temptation, as the infallible preservative from all the powers of darkness. The cross is adored with all the homage due only to the Most High.

Yet, there is much debate about what the cross looked like in the first place and where did this symbol originate. Some believe Christ was crucified on a tree while others hold to the T,t, X shapes wooden constructions. That which is now called the Christian cross was originally no Christian emblem at all, but was the mystic Tau of the Chaldeans and the Egyptians. The original form of the letter T - the initial of Tammuz- which in Hebrew, radically the same as ancient Chaldee, was found on coins. It was called the 'Crux Ansata' or sign of life, borne by Osiris and all the Egyptian gods. They eventually dropped the handle and this became known as the Tau, or ordinary cross as it appears today.

Some Catholics also note the vision of Constantine. But in effect, what he saw was the X which represented the Greek 'CH' for Christ. On a Christian monument to 'Simphonia and her sons' shows the 'X' and not a cross as the heavenly sign saw that day. The Words at the head of the inscription are these: 'In Hoc Vinces X - in this you shall conquer X'

When the Church of God began, there were no symbols for it. Some called it the Way, and eventually a fish symbol arose. It was not until Rome took control that the cross and statues and other 'art' came into focus. Remember

Exodus 20:4-6New International Version (NIV)

4 "You shall not make for yourself an image in the form of anything in heaven above or on the earth beneath or in the waters below. 5 You shall not bow down to them or worship them; for I, the LORD your God, am a jealous God, punishing the children for the sin of the parents to the third and fourth generation of those who hate me, 6 but showing love to a thousand generations of those who love me and keep my commandments.