I have twin boys who have played a lot of board games, and I've made video overviews for some of the games they've played. Here is a video compilation of many of the games they played (by the rules) when they were 3 years old. I've also made a list of 5 of their favorites. Make sure you also check out the
games they played at 2 years old.
Here are links to each individual overview video as well, as well as links to BoardGameGeek (BGG). They are listed in alphabetical order.