Is this orchid dead or dormant?
 

Hi everyone -

I am not a green thumb at all, but was given this tiny orchid for my birthday last month. The blossoms are clearly now dead, but since the leaves are still healthy looking green and seem to be growing continuously, I am wondering if it is just dormant?

If so, where should I cut this? What should I do next?

Thank you in advance for any replies!

If it is “growing continuously“ it is neither dead nor dormant. It’s just through blooming for the time being.

Its alive.
After flowering is a good time to repot the plant if the media is getting really dark / squishy you would want to.
Let the bark dry almost completely out between waterings if you can't tell anything new is growing (roots / leaves).

Dying blooms does not equal dying (or dormant) plant. The plant blooms when it wants to try to reproduce.
The plant looks very healthy and nice. Orchids typically dont go "dormant", except some such as Habenarias, Cypripedium, and other temperate orchids.