IF all travel has been purchased on a single ticket:
Your bag will be checked all the way through to your final destination.
You don't clear customs in Dublin; you clear customs at your final destination. Airport staff at EDI will know just from a glance that your luggage is from outside the EU; luggage checked in at any EU airport has a green-edged label (see below), whereas the label applied by United will have no such green edge, so they will know automatically that your luggage is from outside the EU.
The Common Travel Area (CTA) which somebody introduced upthread has nothing at all to do with customs - and its provisioins don't apply to US passport holders, anyway.
Left: luggage label applied to luggage checked in at an EU airport, recognizable from the green edge. Right: luggage label applied to luggage checked in at an airport outside the EU
IF you are travelling on separate tickets:
Your bag will only be checked as far as Dublin. You will clear both immigration and customs in Dublin (the latter having retrieved your checked luggage), and will then need to make your way to checkin for your onward flight. You must allow SUFFICIENT TIME for any delays or problems that might arise (late arrival, late/non-delivery of luggage, etc) as if you miss the check-in deadline of the second flight when travelling on separate tickets, you lose the value of the ticket and will have to buy a new one at walk-up rates on the day.