Everyone falls. Beginners fall constantly, and good riders fall trying new tricks. It's not a matter of if but how. Fall the right way and you get up, shake it off, and go again. Fall the wrong way and your season is over. Most of the fear that stops people trying new things is really fear of getting hurt, so learning to fall well makes you a braver rider too.
Buy the protection. Helmet, wrist guards, and crash pants (padded shorts with a tailbone pad). It looks like a lot of money when you're just starting out. Compare it to a broken wrist, a concussion, or a back injury that never fully heals. Protection is the cheapest thing you'll buy in snowboarding, and it matters most in your first season, when you fall the most.
The three rules
1. Keep your hands to yourself. The most common snowboard injury is a broken wrist, because your instinct is to catch yourself with your hands. Wrists snap from even a small fall. Butts, hips, backs, and shoulders are far stronger. When you feel yourself going, pull your arms in, bend your knees deep, and sit into it or roll over your shoulder. Wrist guards are there for the times instinct wins.
2. Don't stop suddenly. A fall hurts when all your speed goes into one point at once. A car sliding out on an empty road is fine. The car that hits a wall is not. Sliding out across the slope is usually harmless because the snow takes your speed a bit at a time. Catching your downhill edge, hitting a tree, landing on the knuckle, or getting your board stuck under a rail all stop you dead.
3. Stay in control. The serious accidents are riders going fast along the edge of a run, losing it, and flying into the trees. Ride fast in the middle of the run, not the tree line. Look uphill before you merge. Don't ride drunk.
On the slope
The classic beginner slam is catching the downhill edge: the board stops, you don't, and you go head first into the snow. Keep that edge lifted at all costs. When you do go down, get low first so the drop is short, arms in, and slide out. Then move out of blind spots before you sit and recover, and get back up.
On jumps
Land feet first, even if you fall right after. Landing flat on your back from height is how backs get broken. Get your speed right and land in the actual landing, which is sloped to absorb your speed. The knuckle and the flats are sudden stops. If you wreck, get up and move to the side so nobody lands on you.
On rails
If you feel the fall coming, push away from the rail with your board and fall onto the snow. The bad rail falls are the board stuck under the rail or your head hitting the metal. Helmet on, and detune your edges if you ride a lot of rails.