Best answer by Airgetlam
Well, both positions are correct. Technically, a Play:1 is a streaming speaker, without any way to input anything into it directly, so it needs to be connected to a router. Now, if you have a router, and a computer/phone/NAS, you could then stream your own locally stored digital music to the speaker. But if you wanted to stream anything from the internet (other than Spotify from your phone), you'd be out of luck. Basically, without the internet connection to your router, you've lost almost all of the functionality.
That's likely what the folks on the live chat were trying to say. So, yes, you need a router to hang the Play:1 off of. You can either connect it via local wifi, or connect it directly to the router with an ethernet cable , but it does require a router (i.e. it can't be connected to your computer via the ethernet connection, to my knowledge, unless your computer is set up to be it's own router, which I don't even know is possible).
But without that connection to the whole "world wide web" thing, you won't be able to get the majority of the music that would be available.