"Unicorn" comes from the French and late Latin, with the "cornus" part meaning "horn". I am wondering what other English words share this root. I could think of "rhinoceros". Can you think of something (or multiple somethings) else?
The Latin word for horn is cornu, stem cornu- (with null-inflection in the nominative case). Note that Latin cornus, "cornel/dogwood", comes from a different Proto-Indo-European root and is not related. Rhinoceros comes from Greek keras, horn.
Both Latin cornu and Greek keras come from the same PIE root
*ḱer- (very frequently, and seemingly somewhat at random, expanded to
*ḱerh₂- ), which meant something like "horn, head" (note that there appear to be other PIE roots
*ker- that are not related).
The examples Snumpy and Trideceth gave all come from Latin cornu-. The following words come from the PIE root
*ḱer- through Latin, but not from Latin cornu-:
The word horn comes from the same PIE root, but not through Latin: the /k/ sound was lenited to /x/ or /h/ in Proto-Germanic, as Colin Fine said, which is why we have /h/ now, just as in other Germanic languages.
Previous answers listed cervix, cerebral, triceratops, rhinoceros, cornea, Capricorn, cornicen, corniform, cornucopia, corneous, corner, tricorn, corn (on the foot), and cornicle as being from the same PIE root. There are others, including hirn, keratin, kerato- and anything formed therefrom, cerebro- and anything formed therefrom, many (but not all) other corn- and -corn words, and, possibly, Cornelia. Unrelated are cerul- words, ceremony, karma (related to ceremony, though), hurl, and, as someone else has mentioned, corn (grain).
There are many medical terms as well, which I'll not bother mentioning.
bicorn and tricorn (as in 18th century hats)