На ужин нам принесли по одной порции рыбы, а потом официантка спросила, нужен ли гарнир.

На ужин is an idiomatic way to mean for dinner / at dinnertime.

Russian often uses на + accusative with meals and scheduled events:

• на завтрак — for breakfast
• на обед — for lunch
• на ужин — for dinner

So На ужин нам принесли... means something like For dinner, they brought us...

Нам is the dative case of мы.

Here it marks the people who received something — in English, to us or simply us:

• нам принесли — they brought us / brought to us

Russian very often uses the dative for the recipient of an action.

This is a very common Russian pattern: a past plural verb with no stated subject can mean they in a vague, indefinite sense.

So:

• нам принесли = they brought us / we were brought

The speaker does not care who exactly did it — maybe the staff, maybe a waiter, maybe someone from the kitchen. The person is unimportant.

Also, принесли is perfective, from принести, because it refers to a completed action: the food was brought.

Here по has a distributive meaning: one each, apiece.

So:

• по одной порции = one portion each

In this use, по takes the dative, which is why you get:

• одной — dative feminine singular
• порции — dative singular

This is a very common structure:

• по одному яблоку — one apple each
• по одной книге — one book each
• по две чашки — two cups each

Because один has to agree with the noun it describes.

The noun is порция, which is:

• feminine
• singular
• here in the dative after по

So one must also be feminine singular dative:

• одной порции

Compare:

• один стол — one table
• одна книга — one book
• одной книге — to one book
• по одной порции — one portion each

Because рыбы depends on порции.

Russian often uses genitive after words that mean a container, amount, or portion:

• чашка чая — a cup of tea
• кусок хлеба — a piece of bread
• порция рыбы — a portion of fish

So рыбы is genitive here.

Also, in this sentence рыба means fish as food, not fish (animals) in a general plural sense.

Both are possible in some contexts, but а потом is very natural here.

А does not always mean a strong contrast like English but. It can also mark a transition: and then / and after that / while then.

Here it sounds like:

• first one thing happened,
• and then the next thing happened.

So а потом is a smooth narrative connector.

Ли marks an indirect yes/no question — in English, whether or if.

So:

• официантка спросила, нужен ли гарнир
  = the waitress asked whether a side dish was needed

A direct question would be:

• Нужен гарнир? — Is a side dish needed?

The indirect version becomes:

• ...спросила, нужен ли гарнир

A key point: ли usually comes after the word that is in focus. Here it follows нужен.

Нужен is the short-form adjective used predicatively, meaning is needed.

Russian commonly uses short forms like this:

• нужен — needed (masculine singular)
• нужна — needed (feminine singular)
• нужно — needed (neuter singular)
• нужны — needed (plural)

Here the noun is гарнир, which is masculine singular, so the form is:

• нужен гарнир

Нужный is the full form adjective and usually means something more like necessary / the needed one as a regular descriptive adjective, not the normal predicate in this sentence.

Гарнир is in the nominative.

That is because with нужен, the thing that is needed acts like the grammatical subject:

• Гарнир нужен. — A side dish is needed.

If you want to say to someone, Russian often uses the dative:

• Нам нужен гарнир. — We need a side dish.

In your sentence, нам is omitted in the second clause because it is already understood from the context. The full version could be:

• ...спросила, нужен ли нам гарнир

Because спросила is perfective and refers to a single completed action: she asked once.

• спросила — asked
• спрашивала — was asking / used to ask / asked repeatedly

In a story about a sequence of completed events, perfective is the normal choice:

• they brought the food,
• then the waitress asked something.

So спросила fits the narrative perfectly.