UK, Britain, Great Britain, England, Ireland, Wales... please explain?

I know it should be common knowledge, but I also know a lot of people get it mixed up, including myself, so can someone please explain the denominations, countries, and differences between all of the names of the United Kingdom? Country in itself, part of the state, part of the island...? I really appreciate the help!
Update: What are the accurate descriptions of the states/countries and how are the names used in everyday language (and how are they commonly interchangeable)?
Update 2: Thank you so much for all of the thoughtful answers-- I always want to be politically correct and respectful, so some last few questions: 1) is there a preference/offense for being called english versus british? 2) can someone further explain how part of Ireland is part of the UK, and part isn't? I appreciate... show more Thank you so much for all of the thoughtful answers-- I always want to be politically correct and respectful, so some last few questions: 1) is there a preference/offense for being called english versus british? 2) can someone further explain how part of Ireland is part of the UK, and part isn't?

I appreciate all of the answers and thank everyone for their little tidbits of information. I learned a little something new from everyone. :)
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