Members of the SNP are routinely called nationalists, and the same word is often applied to everybody in the wider pro-independence movement, although Westminster Unionists also like to call us separatists, and international (mainly American) observers occasionally describe us as secessionists.
Of course we're nationalists, but civic ones, which isn't really the primary meaning of "nationalist" in most other countries. This sometimes confuses No campaigners, who at times say things like "I can't vote Yes because I'm an internationalist", although most Yes people have a very international outlook. (In fact I'm often surprised by the number of people in the SNP and in Yes Scotland who have either got family abroad or have lived outwith Scotland for a long time).
Of course we're separatists, insomuch as we want to be ruled by a parliament that is separate from Westminster rather than subordinate to it, but we're very happy to share a lot of laws and institutions with the rUK, with Europe and with the wider world.
Of course we're secessionists to a certain extent, given that it's to be expected the rUK will be more similar to the UK than Scotland will, simply because Scotland is so small in comparison, and because most of the shared institutions are located in London. However, we tend to think of Scottish independence as putting an end to the 1707 Act of Union, which was a treaty uniting two sovereign countries, so we believe we're dissolving a union rather than seceding from it.
Sometimes I just wish people on both sides would agree to call the Yes side sovereigntists, which seems to be the preferred term in Quebec, because that's exactly what we are. The Yes side is united by the belief that Scotland should be a sovereign nation again.
Addendum (11/04/14): Wee Ginger Dug wrote this today: "By the way, it’s far easier to express some political concepts in Spanish than in English. In Spanish you don’t constantly have to have annoying arguments about all independence supporters being nationalists and just the same as Hitler. Spanish has the useful word independentista – which means a person who supports the right to self determination, and nationalism doesn’t come into it. English just has the word “nationalist”. Unfortunately the English version, independentist, makes you sound like a tooth puller for independence, or someone who does freelance fillings."