What is the tense of the following sentence?
They are married.
and what is the part of speech for the word "married"? is it Adjective or Past Verb
Married is a past participle (not a simple past) employed as an adjective. The sentence is in the present tense, marked by the present form are.
Past participles of transitive verbs are employed in passive constructions (BE + past participle), so they very readily come to be used as adjectives describing the state which is imposed by the action of the verb.
For instance, if I lost my keys yesterday, this may be expressed in the passive by saying that my keys were lost yesterday. Today (unless I find them again) I may say that my keys are lost; this is no longer a passive event but a description of my keys: they are in a 'lost' state. And when I do find my keys again I will be able to say that they are found.
Likewise, if two people marry each other we may narrate the event as a passive: Jack and Susan were married yesterday†; but the consequence is that they are now in the 'married' state, and we say that they are married.
† Lambie points out that with marry the passive version is not in present-day English an ordinary transformation of the active: they were not married by each other, but by a third party: "Jack and Susan were married by Rev. Blovious". But the passive does nonetheless give rise to expressing the state they now enjoy as married.
(To) be married, be finished, be dumbstruck
The form: (To) + verb: be + past participle (some people say adjective but that is academic, not practical) is used to describe a person's or thing's state or condition. The past participle (or adjective) is the state or condition a person or thing is in.
We are finished with this book.
We are married now but for a long time we weren't.
The tense of this sentence is present simple as the main verb here is "are". Regarding married's part of speech, it is an adjectival. "Adjectival" is a term used in linguistics, especially in syntax, which means a word that is not an adjective, but that has the adjective position in the sentence.
So the form of "married" is verb, but its position is adjectival so the answer of your question is that "married" is not an adjective like "beautiful" or "amazing" but it works as an adjective in this sentence as it shows the status of the subject "they".