what kind of orchid is it
is it the flower stem that is dead?
then the flower stalk is just done- trim it
is it the stalk below the leaves (dendrobium or other stalked orchid, they sometime get a little corky or tan with age.
leave them be.

if you mean the stem that the blooms were on, it is normal for it to die. The plant hopefully will remain healthy and bloom again next year. Do you know what kind it is?

EDIT: please go to one of the sites listed above and try to identify the type. The only ones I've seen at the WM around here are phaleonopsis and dendrobians. I have had good luck with the former reblooming, but while the latter grows well for me, I have not had them rebloom. Mine get very little attention. Watered every 2 or 3 weeks, fed 4 times a year and 5 of the phaleonopsis are in bloom now. The dennies are not as happy with me. I need to find out if they need more light.

the flower stalk is spent for the season, thats why it looks "dead". trim off the finished stalk or leave it. your choice. continue watering as you always have, assuming it has remained healthy. with luck, it will send up a new flowering stem and bloom again next season. you can also add orchid food to keep it healthy.

if it's the old stem from the flowers when you bought the orchid...it's perfectly normal that it will dry and with patience a new one will start growing with a nice green color. The leaves are still green and alive because your orchid plant is alive and well. Just continue to water as usual and remember to fertilize . With time, you will see a new flower bloom!