I purchased and Downloaded PhotoShop from Amazon for an Apple Computer. I registered it and I was unaware of a "trial run" subscription that has run out and now I can not use my photoshop because adobe keeps opening and saying I don't have subscription. My Question is, do I have to have a subscription to use the program I have purchased?
What version of Photoshop did you purchase? It's strongly recommended NOT to buy Photoshop online, except from Adobe, as there are so many pirated copies on the web.
Adobe stopped selling box/perpetual licensed software at version CS6 ~2 years ago.
If your software is "Creative Cloud" then yes, you would need to log into it with an Adobe Id that has an active subscription.
If you have CS6 or lower you would still be able to activate older software and run it, but it has been out of circulation for a while. A lot of people are getting scammed these days by being sold illegal copies of the volume license seats.
Either way, more info may help getting you up and running.