Gaming is not nearly as stigmatized as it was 10 years ago, but now it has a different issue. People who are not traditional video game players feel overwhelmed by the amount of choices, which adds a barrier for entry. If you want to encourage someone to try out some video games, it is important to show them how casual things can be, and that you do not need a huge amount of dedication or coordination to play them.
Sensory overload is the first issue to overcome when introducing someone to the world of gaming. This can be prevented by showing them games that do not operate in many dimensions. First person shooters are very hard to pick up for a person who has never handled a game controller. The problem is that you have to aim vertically and horizontally at the same time, often while moving, on top of pressing the correct button to fire. That is a lot to take in! Start with simple click based games, or “button mashers” as they are sometimes called. Diablo style games are much easier to start than complex fighting games. Think board game simplicity when making your choice. The first impression matters a lot to the person. Avoid things that have innate time limits, such as timing attacks. Consider it a bonus if you pick a game where they can set down the controller at almost any time and take a break without losing their place. If you are not sure, start more basic, and work from there as you see how they handle each game.
Progression style games may not appeal to average non-gamers because they see it as a waste of time. There has to be some method that feels like they are progressing, even if it is not in the traditional maximized efficiency an experienced player would seek. Skyrim is a good example because the level system is almost secondary, collecting items and exploring will occupy enough attention to keep their interest. Even games where levels are very important can be presented in a fashion so that it is not a big deal. MMO (Massively Multiplayer Online) games like World of Warcraft are often geared to hit the maximum level, and then go fight big bosses. That does not mean these have to be strictly off limits for a newcomer, but it is important to emphasize all the other things that can be done all along the journey. The last thing you want to try and sell someone on video games with is hours and hours of experience grinding.
If you can match their interest with the game style, it takes the “nerd” stigma off the games, and it can become more of a casual activity. A big football fan may find a football game entertaining to play, and their knowledge can transfer over, making them feel at least like they have a head start. That helps develop familiarity with the controls and makes it easier to talk them into trying other games with you. While some may not be interested at all, most people will at least give it a shot.
Video games can be a great source of entertainment; you just have to ease into them. If you grew up playing games as they advanced, you adapted piece by piece. Jumping in now with no experience would be a completely different experience than you realize. Use old fashioned arcades as a guide, maybe suggest a single round of a game while you wait for that pizza to arrive before you start movie night or before the ball game starts just to fill some time. There are video games for all types of personalities and interests; it just takes some trial and error to find the best tool for the job.