The Moshi Monsters game is built using a program called Flash. Unfortunately, official support for Flash is not available on mobile devices like smart phones, iPads, and Android tablets (eg. the Kindle Fire or the Google Nexus 7).
Because of this, it's not really possible to play the Moshi Monsters game on these devices. You can still access other areas of the site, such as the forums and the Daily Growl, but you will be unable to enter Monstro City itself.
The good news is that we do have several dedicated mobile apps already available for Apple devices! You can find them here:
Moshi Monsters Village (Available for Android too!)
Moshi Monsters: Moshlings
Moshi Monsters: Buster's Lost Moshlings
We're also currently busy developing more mobile games for both Apple and Android devices which are due for release soon.