Red Circle on "My trips"
 

What does the red circle with the number1 inside signify above the My Trips square?

You mean this?
https://cimg9.ibsrv.net/gimg/www.fly...503eebb882.jpg

For me, that indicates how many active trips that UA is tracking on my Mileage Plus account: exactly one (1) right now.

David

Generally yes, however there are three cases where it isn't accurate: If you aren't actually logged in to the MP account it seems to be # of active reservations as of last time you logged in; if you've booked a MP trip for someone else (eg when I book a ticket for my wife even if I'm not traveling) that reservation is also included; and within about 24-48 hours after the completion of all segments in a reservation before it 'falls off'

I'm not sure about Android, but as an iPhone/iPad user, that red button UA uses is so synonymous with a 'notification', makes me think something needs urgent attention (flight change/cancellation/check-in notice)...But it's nothing urgent. Was that your initial thought, OP?

And for me as well

I see this red circle on UA.com, either when logged in or when accessing it from a computer on which I was previously logged in, and the number works just as DELee explained. I do not, however, see any red circle on the "My United" button (where my trips can be accessed) in the iOS app.