"The boy sleeps."
Translation:Το αγόρι κοιμάται.
No, the root is κοιμ- followed by a thematic vowel -ά- and an ending -μαι.
The verb has passive endings even though it has an active meaning.
The passive endings in general are -μαι, -σαι, -ται, -μαστε, -στε, -νται.
The conjugation -άμαι etc. in particular, with -ά- before some endings, is comparatively rare, but four common verbs use it, all with active meanings: κοιμάμαι "sleep", λυπάμαι "be sorry", θυμάμαι "remember", φοβάμαι "be afraid of".
The most common passive conjugations are -ομαι, -ιέμαι, -ούμαι which correspond to the three main active conjugations -ω, -άω, -ώ. (For example, γράφομαι, αγαπιέμαι, θεωρούμαι "be written, be loved, be considered" which correspond to γράφω, αγαπάω, θεωρώ "write, love, consider".)
You may recognise the passive endings from most of the forms of the verb "to be" (είμαι, είσαι, είναι, είμαστε, είστε, είναι).