Undergraduate - a type of degree that you do. Usually a Bachelors degree (BA, Bsc). You take this course before you become a graduate, therefore you are an undergraduate doing an undergraduate course.
Postgraduate - University courses that happen after you have graduated from an undergraduate course. These include Masters degrees (MA, MSc) teacher training (PGCE, PGDE) and PhD...
PhD - technically means "Doctor of Philosophy", it is the most common type of Doctorate degree. Often someone's experience at University will be...
Undergrad course > Masters course > PhD > Get a job.
Doctorate - a course to get the title = Dr. Most common one is a PhD
Firm and Insurance choice - Once you have applied to UCAS and received offers, you get to select two of those offers to keep. Your top choice is the firm choice. That's the university you most want to attend out of those who offered you a place. Once you Firm an offer, you are guaranteed a place there if you meet all the conditions of that offer. If you miss the grades on that firm offer - there is your insurance offer. This should be an offer you received on UCAS that has lower grades than your firm choice. If you get lower grades than you firm choice BUT you still meet the offer conditions then you get automatically allocated to your insurance university.