A very limited number of gameplay mechanics:
On a very broad level, the developers of the above-mentioned games have understood what the general mobile game player seeks in a game - what motivates him to play. They know that mobile players tend to be casual players - rather than highly experienced ones. First and foremost, this means that the game has to be easy to learn and understand. Therefore, these games have very few different gameplay mechanics. You tend to do the same action over and over again. This core gameplay mechanic instantly clicks with you because it resonates with something you already know from the real world. Whether it is the logic behind catapults as in Angry Birds, the logic behind rope cutting as in Cut the Rope or the slicing of fruit as in Fruit Ninja, the relationship between what you do, and what plays out on the screen seems intuitive and logical. In other words, these games heavily support the notion of "meaningful play" as explained in this former blog post. When a new feature does crop up, it is gradually introduced and complemented by a short uninterrupting tutorial, clearly showing its underlying logic.
Little input, big output:
In these games, you set something in motion with a simple input, and get a huge amount of feedback usually generated by physics. In Fruit Ninja, a simple swipe generates numerous huge, colorful fruit explosions. In Angry Birds, you catapult your bird with a similarly simple swipe, after which buildings collapse, bricks explode, and pigs fall on each other in a chaotic and rather unpredictable mess.
This relationship between simple inputs and magnificent outputs creates a great feeling of empowerment in players, who instantly see their simple actions have considerable impacts on the game world.
These "considerable impacts" are usually generated by physics. Nearly all successful mobile games are physics-based. They force players to master - or predict - the laws physics in order to achieve success. This approach seems to be clever for a number of reasons. First of all, physics tend to be responsible for numerous emergent gameplay variations, thus providing variety to an otherwise simple game. What's more, the inclusion of physics makes it easier for the player to understand the outcomes of his actions, because he's familiar with the laws of physics from everyday life, thus supporting the notion of "meaningful play", as explained earlier.
A bright, cartoony, and cute visual style:
Both the angry birds in Angry Birds, Omnom in Cut the Rope, or the little bird in Tiny Wings look very expressive and cute. With their big eyes and cartoony outline, they tend to appeal to people of all genders, ages and nationality. As a result, these characters become likeable, and we might even see part of us reflected in them. We immediately create a very basic emotional connection to them. This connection appears to be hugely important - especially for casual gamers. It would be significantly harder to achieve this connection without a cartoony look for a number of reasons. First of all, the small screen size of a phone forces the developer to exaggerate a given expression in order to communicate it. This exaggeration makes sense in a cartoon world. Secondly, the lack of horsepower provided by current mobile devices limits the possibility of creating compelling and credible realistic-looking characters and game worlds.
A very simple storyline:
In order to accentuate the aforementioned emotional connection, a very basic and lighthearted storyline is often set up. Oftentimes at can be summarized with a single sentence. It sets up your character's basic motivation - something irritating or dangerous that needs to be taken care of. Again these storylines tend to have a very universal appeal. Whether bird mothers experience the kidnapping of their eggs, or Omnom needs candy, the story instantly clicks with you - no matter what culture you come from. You understand why the characters do what they do, and you want to help them succeed.
The player is constantly rewarded:
After each play-through of a level, players are rewarded with stars, coins, or highscores or a combination of them, and usually they are accompanied by satisfying and exaggerated sound and graphic effects. This reward-giving act happens extremely frequently in the above-mentioned games. Their developers have understood that for mobile game players, instant gratification is particularly important, as they generally can't muster the amount patience or can't invest the amount of time that traditional hardcore gamers can. These rewards not only provide positive feedback - thus motivating the player to continue playing, they also boost replay value as players revisit levels with sub-optimal scores. Why this constant reward mechanism is generally an important game design principle is further elaborated in this blog post.
A huge amount of levels that are very short...:
It's common for these games to have many bite-sized levels - sometimes over 100. Being bite-sized makes them easy to pick up and play when the player has little play time at his disposal. Sometimes, they take less than 5 seconds to complete. What's more, this structure supports the aforementioned instant gratification that mobile players crave for.
...Or one level played over and over again with random elements:
If these games don't have huge amount of small levels, they tend to adhere to a single level structure as seen in Tiny Wings, JetPack Joyride and Flappy Bird.
Here you play the same level over and over again. Random occurrences serve to differentiate play-throughs, and oftentimes you must complete specific assignments during a play-through. This model is similar to the handheld classic, Tetris, in that the act of surviving for as long as possible in an increasingly difficult game world is a key feature.
2D side-scrolling gameplay:
2D games are seeing a huge revitalization on mobile platforms - not only because the hardware limits complex 3d geometry, but also because 2D games tend to be inherently easier to understand for the casual player.
And the rest...:
Here, I will mention a couple of minor feature enhancements also present in most successful mobile games. The inclusion of leader boards and achievements via services such as OpenFeint and Game Center is one of them. Frequently updating the game with additional content and features is important too. A general high degree of polish, as well as a very low price point - rarely exceeding 3 $ - can also help sales.
As a developer for IOS, I think it can be valuable to look at how many of these features you have included or can include in your game. This post shouldn't be seen as a simple checklist, though. Rather, it's a general guideline for successful mobile game design. It's certainly something that we at Red Key Blue Key have considered greatly in the development of our latest game UFO ZAP.