There are a few ways of downloading music to your iPhone, using either a paid streaming service or a file downloader.
Download Music to your iphone From iTunes & Spotify:
You can download thousands of songs to your phone via Spotify or iTunes, by having a paid account and selecting playlists/albums/etc. for offline use while on your computer. Then, simply open the app on your iPhone.
Download songs or albums straight to your iphone using a File Downloader:
If you want to download specific music files (like mp3 or m4a), you can use Total Downloader.