Update 28/01/2024: As of watchOS 10, the Apple Watch now has a low power mode much like the iPhone. Power save mode is therefore only applicable to anyone running watchOS 9 or older.
You might encounter Apple Watch power save mode when your Apple Watch battery falls below 10%. At this point, a notification appears on screen with an option to enable power save mode. But, it’s not like low power mode on an iPhone.
Low power mode on iPhones just stops your iPhone using up battery in the background by doing things like syncing emails.
Power save mode on an Apple Watch, though, is completely different. It disables all features except the ability for the watch to show you the time. Sadly, it also disables VoiceOver. Because of this, it means that your Apple Watch will basically be useless to you if you enable power save mode.
If you need your Apple Watch later on but not now, the best thing is to power it off to save battery, and then turn it back on again when you need it.