Talking about the geography of the island of Ireland is tricky, because there aren't many fully neutral terms.

Ulster can mean two different things, depending on who is speaking: either the country Northern Ireland (part of the UK) plus three counties in the Republic of Ireland (Cavan, Donegal, Monaghan), or Northern Ireland by itself. The latter meaning is typically used by unionists and can carry the irredentist connotation that Northern Ireland should be larger. The UK doesn't actually claim anything beyond Northern Ireland- it claims six counties out of Ulster, and recognizes the sovereignty of the Republic of Ireland over the remaining three.

Now what I think you're asking is, why does the United Kingdom own part of Ireland (Northern Ireland) and not all of the island? The answer to that is that the UK did control all of Ireland until 1921 (or 1937, depending on who you ask). Starting in the late 12th century, England began taking control of more and more of Ireland, and continually tightened that control until the Acts of Union in 1800 which formed the United Kingdom of Great Britain and Ireland. Uprisings against English/British rule had been occurring pretty much since the English began taking serious control of the island, and finally Ireland fought a successful war of independence that ended in 1921- however, not everyone on the island of Ireland wanted independence. Although it's wrong to characterize the war for independence as a religious conflict, in general the Catholics in the southern part of the island felt "Irish" and wanted their own nation, and the protestants in the northern part (most of whom were descended from colonists who came from Britain in the 1600s and 1700s) felt "British" and wanted to remain in the United Kingdom.

The leaders of the Irish independence movement signed a treaty with the UK which led to independence for most of Ireland, but allowed the six protestant-majority counties in the north to stay in the UK, and that's where Northern Ireland comes from.

tl;dr Ulster is a commonly used (although not completely accurate) term for Northern Ireland, which is the part of Ireland that remained with the UK while the rest of the island became an independent country after 700 years of British rule.