Learn Arabic (Modern Standard): المتحدث حاضر.

This is a classic nominal sentence (a sentence starting with a noun, with no verb). In Arabic, both the subject (المتحدث) and the predicate (حاضر) of a nominal sentence must be in the nominative case. For singular nouns like these, the nominative case is marked by a Damma vowel at the end.

If both words use a Damma, why does one end with an "-u" sound and the other with an "-un" sound?

The exact ending depends on whether the word is definite or indefinite. The subject المتحدث is definite because it starts with ال (al-), so it takes a single short "-u" sound (one Damma). The predicate حاضر is indefinite, so it takes the "-un" sound (a double Damma, known as Tanween).

Where is the word for "is" in this sentence?

There is no word for "is" in this sentence. In Arabic, nominal sentences in the present tense do not use a verb for "to be". Putting a nominative subject and a nominative predicate next to each other automatically creates an equational sentence meaning "A is B".

Is المتحدث related to the verb for speaking?

Yes! It is an active participle, which describes the person doing an action. It literally translates to "the speaker" or "the one who is speaking," which makes it the perfect word for a spokesperson.