Мой будильник звенит утром.

my

Breakdown of Мой будильник звенит утром.

• утром: in the morning
• будильник: the alarm clock
• звенеть: to ring

Questions & Answers about Мой будильник звенит утром.

Why is it мой and not моя or моё?

Мой is the masculine form of мой / моя / моё / мои (my).
It must agree in gender, number, and case with the noun it modifies:

• мой – masculine singular (for words like будильник)
• моя – feminine singular
• моё – neuter singular
• мои – plural

Since будильник is a masculine noun in the nominative singular, you have to use мой будильник.

What case is будильник in, and why?

Будильник is in the nominative singular (basic dictionary form).

In this sentence, будильник is the subject of the verb звенит (it is the alarm that does the ringing), and subjects normally appear in the nominative case in Russian.
So: Мой будильник = My alarm clock (subject).

What exactly does звенит mean? Is it rings or is ringing?

Звенит is the 3rd person singular present tense of звенеть (to ring, to chime, to jingle).

Russian does not have a separate continuous form like English is ringing.
Depending on context, звенит can mean: