Wired or Bluetooth headphones let you listen to music or play games without other people hearing; in-ear or over-ear. For making calls too, you want a [tag:headset] or [tag:hands-free] kit.
Use this tag if your question is about getting headphones to work, or a problem that only happens when headphones are connected. This tag is just as appropriate for large over-ear headphones and in-ear "earphones". Headphones are just listening devices: a single device with a microphone and headphones, for making phone calls, is known as a headset or hands-free kit. With some mobile devices, headphones can also act as an antenna for a fm-radio, so use this tag if you're having a problem getting your headphones to work with an FM radio app.
Shopping recommendations such as "Which headphones are best for..." are off-topic on this site, because they're usually opinion-based and have many correct answers.
This tag is often used with other tags. See also bluetooth for wireless headphones; music, video, or games for things you might want to do with your headphones; music-player, media-player, youtube, google-play-music, winamp, soundcloud, etc. for apps you might want to work with headphones.