Orange is a town in southern France, about 20 km from Avignon.
It's a pretty small place now, but back in the twelfth century it was something of a regional centre. There was a Count of Orange who ran things locally, who by 1163 had become wealthy enough, or powerful enough, or had licked enough arses, to be promoted by the Emperor to become Prince of Orange, ruling over (and collecting taxes from) a small part of what is now France, centred around the town of Orange.
The title was hereditary, and could be inherited by a woman. The family of the Princes of Orange were known as the House of Orange. In the late fourteenh century, the title passed to Princess Mary, who married John, Lord of Chalons-Arlay, another small state. Their son John inherited both Orange and Chalons-Orlay, and thereafter the family was known as the House of Orange-Chalons.
In the 16th century a female member of the family, Claudia, married Henry of Nassau, who ruled a largish chunk of what is now the Netherlands. Nassau was a much larger and more important place and Orange, so she married well. She and Henry had a son, Rene. Her elder brother Philibert was Prince of Orange and, when he died childless, Rene was his heir. Rene thus became Prince of Orange and, when his own father died, Prince of Nassau also. Thereafter the family was known as the House of Orange-Nassau. They continued to live in Nassau, but ruled over Orange as well until the early eighteenth century, when they sold it to France.
William of Orange-Nassau, known to us as William of Orange, was a member of this family. He married Mary Stuart, daughter of King James II of England/VI of Scotland. When James was deposed, William and Mary were invited to take the throne in his place. Thus the House of Orange-Nassau became the ruling house of England and Scotland.
(But not for long, since William and Mary died childless and the throne passed to Mary's sister Anne, a Stuart, and from her to George I, a Hanover.)
Meanwhile the House of Orange-Nassau continued to rule large chunks of the Netherlands and, when the Kingdom of the Netherlands was established in 1815 the first King, William 1, was a member of the House of Orange-Nassau. The House of Orange-Nassau is still the royal house of the Netherlands.
The connection with South Africa is that there were Dutch colonies, and Dutch settlers established there and the name "Orange" was carried there as, e.g. the name of a river, in the name of the Orange Free State, etc, etc. But this happened long after the House of Orange's rather brief involvement in British (and Irish) history, so it's not the foundation for any link between Ireland/NI and South Africa. Any linkage between certain strands of loyalism and Afrikaner politics is simply based on a shared liking for extreme right-wing politics and a shared affinity for racism, and has nothing to do with the House of Orange.