Strategy is all of the skill, thought, talent, planning, and intuition you bring into a situation. Strategy relies on analysis and decision-making, observation and prediction.
Luck is the draw of a card and a throw of the dice (fair dice, that is, not those annoying trick dice your crazy uncle sneaks into games during the holidays sometimes). Even if luck happens inside some boundaries or follows a given probability, it's still luck. You get what you get.
Board game designers love playing with the balance between luck and strategy. Dr. Reiner Knizia, one of the world's most prolific published game designers (and also the creator of many of my favorite games), described how he applies the luck/strategy balance like this: “I like to balance luck and skill. When people finish playing one of my games, I want the winner to say 'it was skill' and the loser to say 'it was luck.'”
Some board games clearly live on one side of the divide or the other. Candyland, Left-Center-Right (the LCR dice game), and Chutes and Ladders are pure luck — draw a card or roll the die. Either way, you're at the mercy of the game. You don't get any choice in the matter; you just do what the game tells you to do. Whether it's the first time you tried the game or the 100th time, your odds or winning are exactly the same.
On the strategy side of the equation, you find traditional games like chess and checkers (or draughts). Every move is calculated, and strategies can stretch over the course of several turns. In these games, the skilled player has a distinct advantage over the beginner, because the experienced player knows so much more.
The majority of modern board games (and many traditional games, too) live in the middle of the continuum, with a mixture of luck and skill. A beginning player might beat an experienced player if the rolls all go the right way, but the experienced player can still put up quite a fight.
From a family perspective, luck and strategy have unique appeals. Games based solely on luck appeal the most to younger kids, thanks to the anticipation and thrill of what might happen next.
Older players typically do better with strategy-oriented games. They can usually think through problems and anticipate moves more clearly than younger kids can, but don't get cocky. Sometimes the younger ones will surprise you with a truly inspired move!
If you're mixing age ranges, a game with more luck might do better than a deeper strategy game. When playing with older kids (or advanced young ones), see how the group feels about playing a 'lighter' luck game or something with 'heavier' strategy.
Be flexible here. If the family wants to play something that isn't your favorite, be the adult and roll with it. Remember that the overall goal is to spend engaging time together. If you're doing that during your game nights, then you already won.