The Apple Watch does not have a headphone port (or a Lighting port for that matter), so wired headphones are out of the equation. But any pair of Bluetooth-enabled headphones will be compatible with the watch.
Pair your headphones
First you need to pair your headphones with the watch - and the way you do this will vary depending on the headphones themselves.
Put the headphones into pairing mode. On most such devices there will be a button to press and hold for a certain period of time - until a light starts flashing, generally. But you should consult the instructions that came with the headphones.
Once they're in pairing mode, turn to the Apple Watch and open the Settings app. (Press the Digital Crown to open the apps page, then tap the cogs icon.) Now select Bluetooth - it's the third option down on Settings' main page.
The watch will tell you that it's searching for Devices (in the top section) and Health Devices (at the bottom), and any that are discovered will be listed in those areas, together with information about whether they are connected to the watch or not. Tap your headphones when they appear.
At this point you may be asked to enter a PIN or password, depending on the security features and settings of your headphones.
There may be a short wait while the pairing process is completed.
Set up playlists
Once your headphones are paired you can stream your favourite songs via Apple Music. But if you want to store music on the watch itself (for listening on a run, for example, if you haven't got the new cellular Series 3 model) then you need to get canny with playlists.
An Apple Watch can store only one playlist of music, so create a playlist on the companion iPhone with all the songs you want to transfer to the watch - call it something like 'Watch playlist' so it's easy to identify.
Now sync the playlist to the watch. Put the watch on its charger, and make sure both the iPhone and the watch have Bluetooth switched on.
Open the Watch app on the companion iPhone and tap My Watch > Music > Synced Music, then tap your watch playlist. A message will appear saying Sync Pending, and then the playlist will start syncing across. Wait for this to complete before taking the watch off its charger and using it.
Play the music on your watch
Now that you've got music on the watch, and you've paired headphones, you're ready to rock. On your Apple Watch, open the Music app, then do a hard-press on the screen and select Source > Apple Watch. Then tap Playlists.