Well, technically, it's neither a raspberry nor a blackberry, it is one of the various species of dewberries found in N. America.

By convention as much as anything, I think people have adopted the term dewberry for the prostrate, vining plants, smaller in stature than blackberries. Some can be very thorny, some not so thorny. Most people consider them a weed, because they can spread pretty aggressively.

Functionally, the fruit is a blackberry, same basic flavor. They are small, but do make good preserves, etc. You can get somewhat better fruit size out of them by fertilizing and watering during the growing season, but yield will always be pretty low on a square foot basis as compared to cultivated and improved strains of blackberries.

Here is a link that might be useful: Dewberries explained via Wikipedia.

The fruit above does not look anything like western native dewberries. What region do you live in, ehooper?

Your plants could also be some seedling berry plant that started growing many years ago, sprouted from one of the nearby berry plants.