An .ipa file is an application for installation on an Apple iOS device, such as an iPhone, iPad, or iPod Touch.
Underneath, an ipa file is actually a ZIP file, so if you want to see what is inside it, you can rename it to .zip and open it with any archiver, such as the excellent 7-Zip.
Just opening an ipa file with the iPhone’s built in browser isn’t enough to install it – the developer must have specified “over the air” installation when building the file (if you’re trying to get this working as a developer, you have to have an enterprise developer account – see more at Stack Overflow).
iTunes used to copy all the ipa files from your device when synchronising, but as of November 2017, Apple announced that’s no longer the case. So in future, when you upgrade your Apple devices, while you can still backup and restore your phone with iTunes to copy all your photos, music and other settings across, you will still need wifi to re-download all the applications. (You can still copy some other things with iTunes, however.)