Are citizens of Northern Ireland considered to be British?

They are British. The full name of the country is the United Kingdom of Great Britain and Northern Ireland. It has its own Assembly and its own government Departments, and in many ways it is semi-detached, but ultimately it is subject to the rulings of the UK Parliament. The Northern Ireland Assembly only has autonomy over matters that are specifically allocated to it, basically relating to local affairs where it is reasonable for Northern Ireland to be different. Northern Ireland is far more semi-detached than Scotland or Wales are - for example, it still issues its own driving licences to Northern Ireland residents whereas Scotland and Wales don't.

Northern Irish politics is still very different from the rest of the UK, though. The major political parties there are all still polarised over unionist or nationalist - wanting to be part of the UK and not wanting to be part of the UK. Traditionally, the major UK political parties don't even stand for election there, with the result that all Northern Irish MPs are always part of Her Majesty's Opposition in the UK Parliament at Westminster. The British Conservative Party has a traditional alliance with the unionist parties, and "Unionist" is still part of its official name, but that's about it. Sinn Fein MPs, being members of perhaps the most extreme nationalist party, don't even sit in the House of Commons as they refuse to swear the required oath of allegiance to the Queen.

The Northern Ireland Assembly is even elected on a completely different basis from anywhere else in the UK. It is elected by Single Transferable Vote as used in the Republic of Ireland. That's just one example of the huge amount of "give and take" there has been to keep Northern Ireland as part of the UK.

It's not short of miraculous that the terrorists on both sides have given up their activities. I was nearly killed by a bomb planted by the IRA (the terrorist branch of Sinn Fein) back in 1983. The whole issue is a complex one and worth researching online if you're interested.

Edit - Andrew H makes a good point. The Republic of Ireland Constitution, until fairly recently, included a territorial claim over Northern Ireland, and the Republic of Ireland still extends Irish citizenship to anyone born there. So the Northern Irish mostly have dual citizenship.

The United Kingdom's full name is actually the United Kingdom of Great Britain and Northern Ireland, so yes Northern Ireland is part of the UK.

All Northern Ireland citizens have UK citizenship, i.e. British citizenship. That's what it'll say on their passports.

However, the Republic Ireland also extends Irish nationality to Northern Irish people if they wish to claim it.

Yes, they are British, it's under British rule but it does have the Northern Irish Assembly, yet it still has to answer to the United Kingdom Parliament

technically they are British

Practically about 45% will not consider themselves as such

Yes there is a Northern Ireland Assembly which has limited powers and has been since the Good Friday agreement. Over the last few years it has occasionally been dissolved because they cant decide whose name comes top on the leterhead and stuff like that - at the moment it is still active.

People of Northern Ireland. Can be either Irish, British or both citizen

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Geographically the human beings from Northern eire can't be British via fact great Britain is an island, which Northern eire isn't area of. yet politically they are British via fact they are area of the union of the united kingdom. i think of its a call between what they might quite be called, or the two.

Yes they are. Most of them or a good many of them are proud of this.

sure, yes they are.