Er online spilling trygt?
As long as you play at reputable, established sites, online gaming is completely safe. Not only is any money you place in your account as safe as it is in nearly any bank, but there's absolutely no chance that the games are rigged or that you're being cheated.
There's really very little difference between online casinos and brick and mortar casinos, in many ways. Security is the top concern for both, as their ultimate success is in making you, the player, feel safe and secure. And, of course, provide a fun environment for you to wager your money in.
Remember, though, that you still need to be very careful with your personal passwords and login information, and protect it at all times. While casinos can protect you from a security point of view, there's nothing they can do to keep information you might store on a computer safe, so a small part of the responsibility for safe online gaming lies with you.