I got a BOSS TU-80 electronic tuner, and have attempted to tune my guitar with it.

Yet when I listen to other guitars online, presumably in-tune guitars (!) they don't sound quite the same tuning.

Is there something about the TU-80 I'm missing?
Some of them allow you to play in other than A-440 (standard) tuning. Make sure it's defaulted to standard. Are your strings in tune if you check the notes the old fashioned way?

Lastly, a lot of songs are not in standard tuning. Instead of changing my guitar, I use the Guitar and Drum Trainer software to shift the song back to standard tuning (you can either just play the songs shifted, or even save them shifted to standard key).

I wouldn't know where to start tuning the 'old fashioned way', as I don't know what that is. However, I have a friend that plays, so I think I'll take mine along and compare the two.
Check your strings against an online tuner. I usually use gieson.com's tuner.
Just got around to doing that - a lot easier than I thought it would be.

Guitar is now in tune with Gieson's...

But now the Boss seems out of tune... at least, the needle is whizzing all the way over to the right hand of the scale.

I think I trust my ear, but the Boss can't possibly be in agreement with Gieson.
I think the Boss should work fine - I used one for years (I use a Snark now!). Do you have the Boss plugged in, or just using the mic? Also, does it seem to be consistently off (like it reads 5 cents too high on all strings) or it's just random?
No, just using the mic.No, it goes way off the scale to the right hand side. It should be centering the needle, yes? And the two lights come on at the same time if it's in tune?
yeah, that's how it should work. Have you tried new batteries? It might be flaky if the battery is low. Also, try plugging it in to a guitar and see if it seems OK that way (that should rule out whether it's able to work - maybe the mic is fowled up).

If you change the tuning, does the needle move up/down in concert with your changes?
This is an acoustic guitar... no leads to play with. I used the batteries it came with, and the display seems bright enough. Maybe I should put brand new batteries in it.

Yes, the needle does change around a bit if I alter the machine head, but I thought I had rad that I would only get accurate readings if the guitar was silenced in between attempts at tuning.

The Giesons tuner is working for me, even if the Boss fails (which I doubt as it is new).