This has come up before, so let's try to keep it brief.
Over the course of history and across the world, black people have been called many things by many people. We had "coloured", and "negro". We also had a series of incredibly offensive names given to us (see any innocuous YouTube comment thread for many examples of these).
Here and now in 2015, we've largely settled on "black" – but we also append what we call ourselves depending on the part of the diaspora we originated in (e.g. "African-American", "Nigerian-British", "Afro-German" etc).
Increasingly, another expression has come to be used as a blanket term for non-white people. That umbrella term is "people of colour". By some accounts the phrase has been around for centuries, but in its current form it is relatively recent, having arrived in the late 1970s. You may find variations depending on context and location: "person(s) of colour", for example, or, in feminist spaces, "women of colour". It has its roots in solidarity, in anti-racist movements where it was politically expedient to combat white supremacy as a bloc.
Note the difference, please. "Coloured" is not equivalent to "people of colour".
A few people have piped up to ask what the difference between "coloured" and "people of colour" is. Here's one Twitter user's eloquent and succinct explanation:
Truly, this is not difficult to grasp.
For further elucidation, here is Loretta Ross of SisterSong explaining the political and ideological 1977 origins of the term "women of colour" in 2011:
Here's an excerpt:
They didn't see it as a biological designation ... It's a solidarity definition, a commitment to work in collaboration with other oppressed women of colour.
And here's another pertinent bit:
We self-named ourselves. This is a term that has a lot of power for us.