The Triumphant Cross is a cross atop an orb. The cross represents Christianity and the orb (often with an equatorial band) represents the world. It symbolises Christ's triumph over the world, and prominent in images of Christ as Salvator Mundi - the Saviour of the World. (See also Earth Symbol.)
The symbol is used by the Carthusian monks1, with the motto Stat crux dum volvitur orbis ('The cross is steady while the world is turning.') The cross and orb have also been adopted by Methodists.
When used as royal regalia, it takes the Latin name Globus Cruciger and is often encrusted with jewels (see also Jewelled Cross). The holder of the Globus Cruciger shows he or she takes responsibility as the titular head of the state religion and is defender of the faith. The orb (or the absurd 'Holy Hand Grenade' in Pythonesque) is also a great way to symbolise a monarch ominously holding an empire.
Before Christianity, Pagan rulers would hold a similar orb in their hands to symbolise their power over the world. When Christian rulers adopted this practice, they added the cross to show Christ's dominion over the world. In this way, it symbolises Christianity's triumph over Paganism.2
And if you fancy a puzzle, see if you can explain this similar curious symbol.
Note: There is no connection between the Triumphant Cross and the identical symbol used in alchemy for the element antimony.