A vízvezeték-szerelő elmondja az eljárást.

Yes. A and az are just two forms of the Hungarian definite article, so both mean the.

Hungarian uses:

• a before a consonant sound
• az before a vowel sound

So here you get:

• a vízvezeték-szerelő
• az eljárást

A useful warning for English speakers: Hungarian a does not mean English a/an here. The Hungarian indefinite article is egy.

It is a compound noun. It is built from smaller parts:

• víz = water
• vezeték = pipe, line, conduit
• szerelő = mechanic, fitter, repair person, installer

So the whole word literally suggests something like water-pipe repairer/installer, which is why it means plumber.

Hungarian often forms long job titles and technical nouns as compounds, and long multi-part compounds are often written with a hyphen like this.

The -t marks the direct object. In other words, it shows what is being explained or told.

• eljárás = procedure
• eljárást = procedure, as the thing affected by the verb

So in this sentence, az eljárást is the object of elmondja.

This is the Hungarian accusative ending, and it is extremely common.

Because Hungarian has two main verb conjugation patterns:

• indefinite conjugation
• definite conjugation

Here the object is definite: az eljárást = the procedure.
Because the verb has a definite object, Hungarian uses the definite form:

• elmond = he/she tells, explains something / tells a story / tells something indefinite
• elmondja = he/she tells/explains it / tells the specific thing

So elmondja matches the definite object az eljárást.

A very useful comparison:

• Elmond egy eljárást. = He explains a procedure.
• Elmondja az eljárást. = He explains the procedure.

The basic verb is mond = say.

With the prefix el-, the meaning often becomes closer to:

• tell
• recount
• say through
• explain fully
• state

So elmondja is more natural here than plain mondja.

In many contexts:

• mond = say
• elmond = tell, recite, explain, relate

The prefix does not always translate neatly into one English word, but it often gives a sense of the action being carried through or completed.

It is:

• 3rd person singular
• present tense
• definite conjugation

So it means he/she/it tells/explains, or with the subject expressed, the plumber tells/explains.

Also, Hungarian present tense is broader than English present tense. Depending on context, elmondja can mean:

• explains
• is explaining
• sometimes even will explain

Hungarian does not have a separate everyday verb form that works exactly like English is explaining.

The sentence as given is the neutral, unmarked order:

• A vízvezeték-szerelő = topic/subject
• elmondja = verb
• az eljárást = object

But Hungarian word order is quite flexible, and changing it usually changes the emphasis.

For example:

• A vízvezeték-szerelő elmondja az eljárást.
  Neutral: The plumber explains the procedure.

• Az eljárást mondja el a vízvezeték-szerelő.
  Focus on the procedure.

• A vízvezeték-szerelő mondja el az eljárást.
  Focus on the plumber.

So yes, you can move things around, but it is not random; it affects what the speaker is highlighting.

In a neutral positive sentence, Hungarian verbal prefixes usually stay attached to the verb.

So:

• elmondja

But when there is:

• negation
• focus
• certain other constructions

the prefix often separates:

• Nem mondja el az eljárást. = The plumber does not explain the procedure.
• Az eljárást mondja el. = It is the procedure that he/she explains.

So the one-word form elmondja is exactly what you expect in the neutral sentence you were given.

A few very useful reading rules:

• Stress is always on the first syllable of each word.
• sz sounds like English s
• s sounds like English sh
• j sounds like English y
• accents such as á, é, ő mark vowel quality/length; they are not stress marks

Rough help:

• vízvezeték-szerelő ≈ VEEZ-veh-zeh-tayk SEH-reh-loe
• elmondja ≈ EL-mon-dya
• eljárást ≈ EL-yaa-raasht

And a few details from this sentence:

• víz sounds like veez
• szerelő begins with sz, so it starts with an s sound, not sh
• eljárást ends with -st, but since s in Hungarian is sh, that final part sounds like -sht

If you want, I can also break the whole sentence down word by word with a very literal gloss.