You can turn on notifications in the settings app for most twitter apps. This works fine for informing you of updates to your twitter timeline on an iPhone or ipad with a data connection. The mobile notification that Musty referred to works by sending you text message on your iphone. If the people you are following tweet a lot, then the text messages may become annoying AND they count against your monthly texting plan unless you have unlimited texts.
I recommend using the notifications feature in the settings app of the iphone and selecting the twitter app you have installed; turn notifications on and select your alert style.