For me, a good game will involve a mixture of both chance and strategy- enough chance to level things out among unevenly matched players, and enough strategy to make it fun to play.
I've made a list of my top ten games. It could easily have been my top fifteen, but I've been ruthless.
1) Settlers of Catan
This is my favourite. It is a building and expansion game, which requires some collaboration, but with plenty of opportunities for being competitive. Even when it isn't your turn, there is still plenty to do, so it is lots of fun to play. There are various expansions which add extra rules. Cities and Knights is the one I like best, but the children generally prefer the Seafarers expansion as they enjoy the different set ups you can make. Probably for 7+, or 6 at a push for child with good concentration.
Another building game, but using tiles to make up a board as you go. The basic rules are easy to follow, but you can get better at it the more you play and the more you learn good tactics. A 5 year old can probably play with help.
You move around the board fighting for garrisons in order to become Caesar, answering questions about Ancient Rome on the way. The questions are multiple choice, and you can choose easier or harder ones, so even young players can enjoy it. Probably 7+.
4) Monopoly Deal
This is a card game based on the big game, but much quicker to play (about 15 minutes a round). I slightly prefer it to Monopoly. It is a good choice for taking on holiday. The rules seem a bit confusing at first, but it's actually fairly easy to play. 6+ for a competent reader.
A classic dice game, which also doesn't take too long for a round. It's good for practising maths too. 5+ with a bit of help.
6) Mexican Train
This is a new addition to our collection, but has been popular. It is basically elaborate dominoes. You can play a long game, a short game, or keep score over a number of days. There are plenty of opportunities for using good strategy, but my 3 year old was able to play on her own with only a little help (though not to win!). 3+.
Another classic game, and one which is great for encouraging logical thought. Once again, this is a game that can be played at a simple level, but which you can do much better at as your deductive skills increase. Probably 7+.
This is our game of choice to play with adults, usually at a Bible study group social. It isn't too intimidating if you make it clear that you can join a team in a merely guessing capacity. It gets people chatting and relaxing together, and bonding over a shared ignorance of famous people and geography. I haven't actually played this competitively with our children yet, though I have tried it out with my older two (they loved it). It says 12+ on the box (seems right to me).
A simple memory game. One of the few games for very young children that everyone is happy to play! The children usually beat me at this. 2+.
A simple card game which is also good for all ages. Anyone who can hold and manage their cards can play. Probably a mature 3+ (perhaps with somewhere to spread their cards out).
Note- there is not a single "educational" game on my list. The children have enjoyed them, but I would generally prefer to eat my own arm rather than to play them. They usually involve no strategy whatsoever, and take an age to play (or it feels like it!). So we play real games together, and have fun while incidentally learning stuff like arithmetic or logic, or even how to lose with reasonable grace.