LittleToe did a good job explaining everything. However, I realise it's a lot to take in all at once. To make it simple for you the situation at a very basic level is this:
Scotland is a country
England is a country
Wales is a country
Together, these 3 countries make another bigger country called Great Britain. This is the island in western Europe.
You can only be English if you come from England. Scottish and Welsh people are not English.
Scottish, English and Welsh people are all British.
Great Britain plus Northern Ireland plus every other territory that remains of our dead empire (i.e Falkland Islands, etc) makes The United Kingdom. For example...if I was from the island of Montserrat in the Carribean...I would be part of the UK, but not British, Great Britain is the island in Europe remember.
Finally, Northern Ireland is technically part of the UK (Like Montserrat, the Falklands, etc) and not British... however, for all intents and purposes, they are Brits. Why this is I am not sure. Maybe someone from N.I can correct me if I'm wrong. Hope this helps!