A nappaliban olvasok, amíg a barátom tévét néz.

The base noun is nappali = living room.
The ending -ban / -ben is the inessive case, meaning “in” a place.

• nappali → living room
• nappali + ban → nappaliban = in the living room

Hungarian often uses a case ending instead of a separate preposition like in. So A nappaliban literally means “In the living room”.

A is the definite article (“the”).

• A nappaliban = in the living room (a specific one both speakers know about)
• Nappaliban without A would sound incomplete or odd here; with common indoor locations like a konyhában, a nappaliban, etc., the article is normally used when you mean a specific place.

So in this sentence the article A is natural and normally should not be dropped.

Hungarian usually drops subject pronouns when they’re clear from the verb ending.

• olvasok ends in -ok, which is the 1st person singular indefinite ending. That already tells you the subject is “I”.
• Én olvasok is also correct but adds emphasis to “I” (like “I am the one reading”).

So olvasok by itself means “I read / I am reading”.

Hungarian has one present tense for both English simple present and present continuous.

• olvasok can mean:
  • I read (in general, as a habit)
  • I am reading (right now)

Which one is meant comes from context or time expressions.
Here, with amíg (“while”), the meaning is clearly “I am reading” at that time.

In this sentence, amíg means “while” or “as long as”:

• A nappaliban olvasok, amíg a barátom tévét néz.
  → I’m reading in the living room while my friend is watching TV.

About amíg vs míg:

• In everyday speech, they are largely interchangeable, and both can mean “while” or “until”, depending on context.
• Some style guides suggest subtle differences (e.g. amíg more “as long as”, míg more simple “while”), but for a learner it’s safe to treat them as practically the same here.

Hungarian uses a comma to separate a main clause from a subordinate clause introduced by conjunctions such as hogy, mert, amikor, amíg, etc.

• Main clause: A nappaliban olvasok
• Subordinate clause: amíg a barátom tévét néz

So the comma before amíg is required by Hungarian punctuation rules, similar to English before “while” when it starts a subordinate clause.

The base noun is barát = friend (often boyfriend in context).

The ending -om is a possessive suffix meaning “my”:

• barát → friend
• barátom → my friend / my boyfriend

So a barátom = “my friend” (definite article a + possessed noun).
Hungarian usually uses these suffixes instead of a separate my/your/his word.

Literally, barátom = my friend.

However, in everyday speech:

• barátom (said by a woman) often implies my boyfriend.
• barátnőm (from barátnő) = my (female) friend or my girlfriend, depending on context.

If you want to be unambiguously romantic:

• a barátom (for a male partner)
• a barátnőm (for a female partner)

tévé is the base noun = TV.

Hungarian marks the direct object with the accusative ending -t:

• tévé → tévé + t → tévét = TV as a direct object (watching TV).

So tévét néz literally means “watches TV”.

This is a subtle but important difference:

• tévét néz = he/she is watching TV (in general)
  → no article, indefinite object, more like an activity: is watching TV.
• a tévét nézi = he/she is watching the TV (a specific set)
  → article a
  • definite object, and the verb form changes (see next question).

In the sentence, the idea is “is watching TV” as an activity, so tévét néz (no article) is the natural form.

Hungarian has indefinite and definite verb conjugations:

• néz = indefinite (no specific definite object, or no object at all)
• nézi = definite (there is a grammatically definite object: a tévét, azt, őt, etc.)

Compare:

• A barátom tévét néz.
  → My friend is watching TV (in general).
  → Object tévét is indefinite, so the verb is néz.
• A barátom a tévét nézi.
  → My friend is watching the TV (set).
  → Object a tévét is definite, so the verb is nézi.

In your sentence, we’re talking about the activity, so tévét néz is correct.

Yes, that word order is also grammatical:

• A nappaliban olvasok
• Olvasok a nappaliban

Both mean roughly the same: I’m reading in the living room.

However, Hungarian word order is used for emphasis:

• A nappaliban olvasok slightly emphasizes the place (“It’s in the living room that I’m reading”).
• Olvasok a nappaliban is more neutral, or can emphasize olvasok in some contexts.

Similarly, in the second clause you could say:

• a barátom tévét néz (neutral)
• a barátom néz tévét (more focus on the verb néz)

For a learner, both orders you suggested are acceptable; the original just gives a bit more emphasis to the locations.

Yes, tévét néz is a very common, almost formula-like expression:

• tévét néz = to watch TV

People also say:

• tévézni (a verb formed from tévé) = to watch TV (informal)

So your sentence could be rephrased informally as:

• A nappaliban olvasok, amíg a barátom tévézik.
  → same meaning, just more colloquial.