All my kids started playing board games around their 2nd birthday. Of course, we often modified the rules and were more flexible to make them more age appropriate. Or they'd sit next to an adult or older sibling and 'help' them by moving pieces, drawing cards, etc. This got them used to the idea of taking turns in a game.
My youngest is 2.5. Her current favorite games are Memory (we don't use every card - we use about half the cards to make it easier for her), Chutes & Ladders, CandyLand, Guess Who and Zingo. She'll often partner with someone when we play Rummikub, Uno or Phase10 and is actually getting good at finding runs and such. Last night my 7 year old taught my 4 year old how to play Risk and the 2 year old was right there listening and attempting to play with them.
There are tons of games out there today marketed for toddlers. Just be sure they are open ended and actually force the child to use their brain - too many are close ended and 'dumbed down' for kids, which is insulting. As many others have said, games are essential for kids to play because they teach many social skills -- losing and winning gracefully, taking turns, thinking 2+ steps ahead, math, reading, etc.