Yes those are black raspberries. Berries fruit on second year growth (floricanes). Floricanes die after they fruit so you can remove them then or in late winter next year. Cut them at the soil line.

Primocanes (this year's new growth) should be topped by removing about 6 inches once they reach about 2-3 feet. This will encourage lateral growth and will give you more fruit for next year.

In late winter, prune the canes again (they are now floricanes) remove weak spindly canes, cut canes back to about 4-5 feet in height and cut back lateral growth to 6-8 inches. Canes will need to be supported by tying them up to a wire trellis or staked.

Thank you for the very informative post.

The black raspberries pictured above are not fully ripe -- a lot or reddish areas are showing. Black-caps are fully ripe when they will pull off very easily. So easily that the rest of the plant barely moves. The time difference between a firm pull (unripe) and an easy pull is about 2 days. If you shake a whole cane and no berries fall, hardly any berries will be fully ripe.

Thank you.

In my yard they won't last until completely ripe. The birds will attack them as soon as they get mostly dark.