Duerme - he or she sleeps. So what about 'dormir'? Which by the 'ir' I am guessing indicates the verb 'to sleep'. Just that I have never seen 'duerme' before.
"Dormir" is an irregular verb, like mexicanfoodfreak and lesliedmckenzie were talking about above. The verb "dormir" is a stem-changer in both present and preterit (past) tenses. In present tense, the "o" is changed to a "ue" to keep the sound consistent. Hope this helps!