Yeah, as far as I'm aware, Northern Ireland is part of the UK. The United Kingdom of Great Britain AND NI. The two constitute that United Kingdom, whereas Great Britain comprises England, Scotland, and Wales.
It's confusing because the same word is used to mean different things depending on context. This is due partly to linguistic laziness and partly to the complicated history of the British Isles (its self a contentious term.)
The country is called "The United Kingdom of Great Britain and Northern Ireland" and Northern Ireland is definitely a part of it. The problem is that the full name is a bit of a mouthful to say, a problem shared by the USA but for some reason UKoGBaNI never caught on. So we need a short version - the current accepted shortening is "The United Kingdom" or UK for short - easy.
Now the problems start, in ye olde days it was quite common to use "Great Britain" as a short version of UKoGBaNI. In fact the "official" internet identifier for the UK is .gb. The problem with this is that Great Britain is also a geographic term, it's the name of a big island of the coast of Europe. England, Scotland and Wales are physically on Great Britain and NI isn't (the location of NI also has room for linguistic confusion, I'll come to that later.) So if someone talks about Great Britain what do they mean, UKoGBaNI or the big island? Very difficult to tell from context as there is so much overlap, 90-odd% of UKoGBaNI is on the Island of Great Britain so even if people say the wrong thing most of the time they will be correct by default.
Other potentially confusing terms:
Ireland. The smaller of the British Isles is called Ireland, this is a geographic term. There are two countries on the island, the province of NI (which is part of UKoGBaNI) and the country most of us know as "The Republic of Ireland." This isn't its name though, it the official description, the countries name as defined in its constitution is "Ireland." More room for confusion, NI is part of Ireland (the island) but not part of Ireland (the country.)
Britain. This has no independent meaning. It can either be a short form of UKoGBaNI or Great Britain, no way of telling easily from context.
British. Probably the easiest term here, the accepted word to describe a citizen of the UKoGBaNI. Can be contentious in NI.
Irish. Back to confusion. Citizens of the country (the Republic of) Ireland are Irish citizens. Irish is also an identifier that can applied to anyone from the island of Ireland, this included people from NI, who are British. To complicate things the RoI extends Irish citizenship to anyone born on the island of Ireland, which of course included NI. So most people in NI are British, Irish (as in come from the island) and Irish (as in citizen of RoI.) Only a lot of people in NI, depending on their political bent, will reject one or more of those identifiers.