In general, downloading can mean getting files from any remote computer or server to a local computer. On this site, downloading has come to be used more specifically for transfers to the Android device over a network (whether that's a local Wi-Fi network or the Internet).
Use this tag whether you're downloading applications, files, or other data, and whether you're using a browser or a special-purpose app. For questions relating to downloading within an app that has wider functionality, such as google-play or dropbox, please include this tag alongside the tag for the app. If the question's specifically about the Downloads system app, use download-manager.
If you're interested in installing apps onto your phone from a PC (instead of an app market), the relevant tag is side-loading. If you're installing a new ROM or firmware onto your device, use the rom-flashing tag. Even though this process can be called downloading and may use the download mode of your device, it would be confusing to use the same tag for downloading things over the Internet as well as installing ROMs.