This thread has been widely discussed on a lot of forums. To make a long story short, running natively R on an iDevice (meaning iPhone/iPad) is disabled by its OS, unless it is jailbroken. The steps for the installation through Cydia are described in this R wiki, or this post. But there are some limitations, including bugs in package management. Running R directly seems not to be the more sensible way to proceed: smartphones lack so much power for that, you would do it just for fun. Typing R code on a small keyboard is quite tedious. But is manageable on a larger one (iPad, Galaxy Tab).
There could be another short-term solution to use R on an iDevice (or any other smartphone). As explained on this introduction to RStudio, the novel IDE can be launched directly on the Web, while running on a server. This would be an easier and safer solution than jailbreaking the mobile, which would have a number of benefits, including: the ability to access and share your files from anywhere (with an Internet connection), to benefit from more powerful computing resources (of your server), and to focus on a single configuration of R (for packages).
But there is a but, because the RStudio development team is still working on it. It is close to working (it is possible to click on tabs and menus) but not yet possible to get the keyboard to initialize and to type in the console. Yet executing the code can be done by clicking on Run line(s)+Enter. It is not a lot for now, but is a good start.