There are over 1,000 apps submitted to the App Store every day. With all of these apps, making a game that stands out is pretty difficult. And making a game that remains popular over a larger period of time is even harder.
Now mobile games pretty much fall into one of five categories.
RTS - These games are real time strategy games. You are playing in multiplayer, there are constantly improvements to your base and troops, and you are at risk of being attacked. Games like this include Valor, Clash of Clans, Game of War, and more.
Endless Levels - These games combine the endless and the leveled categories. These games are technically split into discrete chunks, and there are a lot of them, but either the chunks are timed, or limited by practical skill that they're not truly endless. Games like this include Candy Crush, ZombieVille, Dots, every Bejeweled clone
Now obviously I'm cherry picking the examples from the winners in these fields, but it shows that most successful apps (and a lot of unsuccessful apps) come from one of these five categories. Each category does well for slightly different reason.
Endless games do well because of the near-miss phenomenon. Basically, because the games are endless, it's incredibly easy to restart (sometimes even done automatically), and it's a small and simple mistakes that kill you, people think that if they just play "one more time", they'll do better. This is the same phenomenon of why slot machines are programmed to show near-misses more often than chance (two of a kind and one other). This leads to the constant play that made Impossible Road, Flappy Bird, and Subway Surfers such popular games.
Leveled games do well because they're just that: leveled. A mobile game is usually going to be played in short bursts, when there's down time in whatever activity they're actually doing. Most levels should fit neatly into a <5 minute time window. Most are considerably less. Additionally, the key to a leveled game is a slow but continuous build of concepts. The leveling structure allows you to easily build on the core concepts of the game, and turn what first is a simple one into a more and more complicated one. This follows the idea of "Bushnell's Law", which basically says that games should be easy to learn and difficult to master. This further cultivates the addiction, which is (arguably) what you want in a mobile game.
RTS games are similar to leveled games. By making the options for what you can do limited at a low level, and then slowly increasing it as you gain access to more stuff, it continues to follow Bushnell's law. Additionally, because it's time based it can also just be fiddled with in moments of down time. In a system of rewards, there are four schedules: fixed ratio (rewarded after a number of things), variable ratio (rewarded on average after a number of things, but variable), fixed interval (rewarded after amount of time), and variable interval (rewarded on average after amount of time, but variable). The most addictive schedule is variable ratio, as you constantly have to do some task as opposed to wait, and there's an element of chance which sucks people in. RTS' are the second most addictive, which is variable interval. It demands a large amount of time put into it, as you're not entirely sure when things are going to start paying off.
Endless leveled games are a combination of the endless and the levels. They have the more open-ended continually simplistic nature of the endless, but the time constraint of the level. These games are addictive because walls are put up. These ensure that you keep checking in, by giving time based rewards (being able to keep going through levels, or recharging some power up). They're also addicted because of the same near-miss phenomenon that drives endless games.
Personal competition games are fun simply because of a desire to compete with your friends. Whether it's in opposition, with Words with Friends, collaborative like in Draw Something, the personal connection is stronger in the 1v1 setting.
There's one thing I haven't touched on though, and that's monetizing these games. Check out part 2 for that.