If you’re downloading a single media file, this might not be as easy as you'd expect, because iOS doesn't handle single media file downloads correctly. Instead of saving the file, it will stream it in your mobile browser or open it in an app designed to open these files directly. You can solve this by asking the sender to compress the file into a .zip file (or if they send multiple files, download it as a zip), so your phone will have to download it first (make sure you have WinZip installed) and then you should be able to save it to your phone/tablet.
If it is an image however, when it opens in your browser, you can also press and hold down on the screen over the image and the option to save the image will appear.
Otherwise, it might be an idea to download the files via a desktop Mac or PC and then transfer them via iTunes to your phone.