Yes, you can log into TurboTax Online with the same account via multiple platforms - i.e desktop computer, ipad, iphone. You just need to make sure you are logging into the account with the same user ID and password. Your information gets stored online as you work on it and you can access it from any device.
There is also a CD/Download version of TurboTax (you will see it sold at stores like Costco or Amazon). This version you need to install onto your computer and you can only work from your computer. So, if you want to be able to access from multiple devices, then you need to use TurboTax Online.