Most backpackers work their asses off before their big trip. Some work 45-60 hours a week for 4-6 months while others take their time and plan a couple of years ahead working at the local supermarket every Saturday morning. Either way, if you want to save the dough for your trip, make sure to let go of expensive habits such as shopping, eating out and partying hard - things you will do plenty of once you are on the go.
You have other options. One is obtaining a loan from your relatives or a financial institution. It's an easy option, but think again - do you really want to start out your gap year and (perhaps) future education with a fat loan hanging over your head?
Another solution is to work your way around the world. If you're good at keeping income and expenses on a level, this may work for you. A popular concept for backpackers with plenty of time to spare is to enroll in a Working Holiday program, usually in Australia, New Zealand or Canada. This entitles you to a Working Holiday visa in your passport with a right to legal work up to 6 months (or more) out of a year. Apart from the fact that you earn some money, it's an easy way into exploring local culture and socializing with other Working Holiday makers and the local population.