Scots-Irish is a very confusing term, but it actually has a real meaning.

First, it does not refer to someone of mixed Scottish and Irish heritage. It is incorrect to use it in this manner.

It does refer to a particular migration pattern, which began in the Lowlands of Scotland in the early seventeenth century, moved to Ulster in northern Ireland, and then during the eighteenth century migrated to North American (largely to Pennsylvania, though to other port areas, as well).

I have an article about the migration here:
http://www.scottishtartans.org/ulster.html

In brief, these were people of Scottish descent who had been living in Ireland for a time prior to their arrival in North America. Think of it as akin to the term Scottish-American (people of Scottish descent living in America). One other thing to keep in mind is that "Scots-Irish" is largely an American term. The people were not called that until their arrival in North America, largely to distinguish them from the Irish Catholics, who arrived in another migration entirely.

Those of Scottish descent who still reside in northern Ireland are usually referred to as "Ulster Scots."

So, unless you can trace your heritage back to the migration of the Ulster Scots to North America, it's not really correct to refer to yourself as "Scots-Irish."

Aye,
Matt