Providing you behave responsibly and avoid putting yourself in danger by taking too many risks, skiing and boarding are relatively safe sports - leading to fewer injuries per head than such run-of-the-mill sports as keep-fit, tennis, squash, cricket and rugby. So, as long as you are sensible you should be confident in the knowledge that you are not putting yourself unduly at risk.
When on-piste make sure to follow the rules of the 'Skier's Highway Code' (covered in 'The Rules' section, which are designed to avoid collisions and promote enjoyable, safe skiing and boarding.
On-piste you should be safe from most of the natural dangers of the mountain. Your biggest risks are falling on ice or colliding with a tree or with another skier or boarder due to your own or their incompetence or recklessness - so make sure you watch and keep a safe distance from the people in front (downhill) of you.
Remember to stay within the bounds of the piste - marked with sticks or posts. (How well they are marked - which matters during a white-out - varies a lot between resorts, but in principle there is a clear definition of where it is safe to ski and board.)
Read your piste map carefully to avoid venturing into territory that is beyond your capability or you could be getting yourself into sticky situations. The runs are graded for difficulty so you should be able to work out what you are capable of doing. Remember though, grading can be very inconsistent between resorts - some Val d'Isère greens would be graded red in many other resorts, for example - so if you are a novice, treat unfamiliar runs with some caution.
Pistes are checked against avalanche danger; if there is a danger, they are kept closed until the avalanche has been artificially triggered, or the snow has stabilised naturally. They can also be closed due to lack of snow, which might uncover rocky patches. So don't venture under the barriers of a closed piste - you will be putting yourself and perhaps others at risk.
In general, pistes are patrolled so that injured skiers and boarders will be found - especially at the end of the day when the runs are checked after they have been closed. (Patrolling should also mean that obstacles such as rocks and bare earth are marked.) If you are injured, wait calmly for help.
But be aware that in at least one area in the Alps - the Jungfrau region shared between Wengen and Grindelwald - the end-of-day patrol does not cover all runs. And be aware more generally that not all resorts rope off runs once they have been patrolled at the end of the day - so at the end of the day, unless you are careful, you can find yourself skiing a run alone, without the reassurance that a patroller is following behind.
For more ski safety, tips and mountain advice: Ski lifts | Pistes | Snow | Accidents and First Aid | Altitude sickness | Avalanches | Ski Guiding | Sunburn and snowblindness | ISF rules | Weather | Off-piste safety