Black
People with roots in Africa or the Caribbean generally prefer this word to describe themselves (though some older people may not). Of course they are not really black like shoes can be black, but then 'white' people are not really white, are they? One of the reasons the word 'black' is preferred is that in the past people were often taught that black = bad or evil, and many people now want to say that there is nothing bad or evil about dark skin and that they are proud of it. As long ago as the 1960s black people in the USA summed this up in the phrase 'black is beautiful'. Some people with Asian roots call themselves 'black' but most don't.
African-Caribbean
This is the term people with roots in the Caribbean tend to prefer, as an alternative to 'black'. They prefer it to what they used to be called, which was 'West Indian'.
Asian
This is the most general word for people with roots or family connections in India, Pakistan, Bangladesh and Sri Lanka. You aren't likely to annoy anyone by using it.
Pakistani, Indian, Bangladeshi, Chinese etc.
If you know someone has roots or family connections in one of these places then one of these words is fine, though people can be touchy if you kind of suggest they are not really British when they think they are. If you had a friend with an Italian name because her Italian grandparents moved to Britain in 1950, would you call her Italian or British? Perhaps you would not be sure, perhaps it might depend on whether she felt a bit Italian herself, spoke Italian, went to an Italian-speaking Catholic church?
Roots or family connections in...
This is a useful expression. Taking India as an example, some people in Britain came here from India in the past few years (not many, actually); some people have been here forty years, others (almost everyone under the age of 25) were born here. In this mixture some have Indian passports, most have British passports and most know no other home other than Britain. They are British, but they have roots or family connections in India.
Ethnic minorities
This is a funny phrase because it's often used in quite a vague way. Actually, you will find that people often use it when they mean black and Asian people, though ethnic has nothing to do with colour. The Irish in Britain are a minority ethnic group. You could say Welsh people are an ethnic group.
Muslim, Sikh, Hindu etc.
Sometimes a person's religion is more important to them than their family's roots, so it is sometimes better to describe them as a Sikh (for example) than as an Indian.
So would you put all of the above into ONE category?