I have been asked recently if I am Christian or Catholic? What kind of question is this, and how should I answer it?

You are right to be annoyed and find offense. Catholics are Christian, indeed the original Christians. We have been here all 2000 years, and are the Church founded by Jesus Christ himself. 
 
It is sadly true, that there are some who polemically and rhetorically ask, “Are you Catholic or Christian?” as if the two categories were mutually exclusive. They are not. It would be like asking a certain man, “Are you male or human?” And of course, the answer is both and the question as stated is offensive. And the same answer is true here, that Catholics are also Christians. 
 
Unfortunately, not all Christians are Catholic. And this is a countersign, because Christ founded one Church, and has one Body, he prayed for unity, not endless divisions. He did not found tens of thousands of different disputing denominations. He established one Church which he founded, and unified around his vicar, Peter and his successors, the popes who are designated to unify and strengthen the others whom the devil would sift (separate) like wheat (cf Luke 22:31). 
 
The word “Catholic,” while often used as a proper noun, is also an adjective. The word catholic comes from the Greek, and means “according to the whole.” And this is understood in at least two ways. First of all, the Church is Catholic, because we preach the whole counsel of Christ, not just certain favorite passages or popular viewpoints. We are called to preach the whole Gospel, whether in season or out of season. 
 
Secondly, the Church is called catholic, because we are called to a universal outreach to all the nations. We are not just a church of a certain race or nation, we are called to go on to the ends of the earth, and make disciples of all nations. The Church has a “catholic,” that is a universal, mission to everyone. 
 
And thus, the Catholic Church is the Christian Church, and all Christians are called to be Catholic. It is for this unity that we must lovingly strive.