Let me start this post by saying that games are good, especially for children. They are a wholesome social activity in which they can interact with their parents on a more even playing field, and an excellent way to teach any number of skills (reading, math, memory, spatial relations, etc.) and develop their dexterity.
With that said, some games...let's use the phrase "quickly outlive their usefulness." Perhaps this isn't the most correct phrase, but if you're a parent who has just completed their 900th game of Candyland, you understand what I mean.
The goal of this post is to suggest games that serve this purpose, that is, aid the development of one's children, but don't bore a parent into a coma.
First, here's a link that takes you to a small list of games for children of about age two. While not linked here, with a little attention and tweaking, most children around age 3 should be able to play games listed "for ages 5 and up"--this greatly extends the number and variety of boargames available for play. Also keep in mind that development at this age is amazing--a child at 2 years, 2 months will be far more developed than they were at only 2 years.
Now, some more specific suggestions of my own, with summaries. Many of these games take 20 minutes or less to complete.
--Players draw wooden spheres from a small wooden bowl without knocking over the "alarm" stick; player moves to the same color on a track as the sphere they drew. The goal is to get to the end of the track first. Twenty minute play time, and kids tend to excel over adults due to smaller fingers. Note: The wooden spheres look somewhat like peanut M&Ms--if you children are familiar with this candy, either pay really close attention or pass on this game.
Tier auf Tier (Animal upon Animal):
--Each player has a set of thick wooden animals. On their turn, a player rolls two dice, and must stack the associated animals upon the crocodile (or upon the other animals already on the crocodile). First to stack all their animals without knocking over the crock wins. Good, simple dexterity game with solid, sturdy pieces.
Another dexterity game, this game comes with a wooden ring and four sets of colored wooden hippos, six to each set. One player spins the wooden ring; everyone plays simultaneously, flicking their hippos. Players attempt to get the hippos in or under the ring once it stops spinning; these are removed from play, and the ring is spun again to start a new round. The first player to get rid of three hippos wins.
Hey! That's My Fish!:
For two to four players, I'm actually considering this one for a child slightly older than two--perhaps three or four years old. The board is composed of a number of hexagonal tiles, each depicting 1, 2, or 3 fish. Players have two or three wooden penguins, and after placing them on the board, take turns moving any of their penguins as many spaces as desired in a straight line, and remove the tile the penguin started on. Eventually the board develops holes or separates into several pieces (think of the motorcycle things in Tron--hopefully everyone has seen that movie) as the penguins jockey for board position. The player with the most fish at the end of the game wins.
Lastly, I'll make two suggestions that I think are funny and unique.
Run for your Life, Candyman!:
If your kids can play Candyland, they can play this with a little help. This is Candyland in reverse--after discovering that the King of Candyland has been selling the gingerbread people to the children of the world (for eating!), you (a gingerbread person, of course) decide you need to escape. Of course, only one can make it, so if you have to beat on the other gingerbread people, so be it. This game plays exactly like Candyland except for two things: first, you hit and damage any player that you pass--this can ultimately lead to destroying an opponents arms, legs, etc. Second, you get weapons to help you hit your opponent more and/or harder. There is a little reading, but if you are selective with the weapon cards (all of them are cute, candy themed weapons) and play with open hands, this shouldn't be a problem.
Purely a dexterity game, players flick their cars (wooden disks) around a smooth wooden track. The goal, of course, is to be the first to pass the finish line. Great fun and playable by anyone that can flick, it is, unfortunately, on the pricey side, at around $80.
BoardgameGeek.com is an excellent resource for all manner of boardgaming knowledge, has an excellent database of boardgames, and, best of all, it's free to use. All links in this post will likely lead to Boardgamegeek.com.