Major = more or less your specialisation within the broad field covered by your degree. In most degrees you'll have to pick one or two majors. A major involves doing a minimum number of units within that study area.
Master's = a postgraduate degree (as opposed to a bachelor's, which is an undergraduate degree). Typically covers more advanced material and/or involves a major research project. You can't generally do it until you've done a bachelor's degree in the same area.
Honours = generally an extra year added onto a bachelor's degree in which you specialise in just one topic, which you'll need to have majored in during the main part of your degree. It typically involves a major piece of research. You get 'Hons' in brackets after the degree abbreviation, so a B Arts with honours would be BA (Hons).
PhD = a doctorate. (Short for "doctor of philosophy", but doctorates are known as PhDs in all disciplines, not just philosophy.) Involves 3+ years of original research leading to an 80,000-100,000 word thesis. You would typically need either a first-class honours degree or a research master's to proceed to a PhD. Once you get one, you can use the title 'Doctor'. It is more or less the gateway to an academic career.