La prezo de la nova komputilo gravas por mi.

The first la goes with prezo: la prezo = the price.

The second la goes with nova komputilo: la nova komputilo = the new computer.

So Esperanto is being very literal here:

• la prezo = the price
• de la nova komputilo = of the new computer

English sometimes uses fewer articles in similar phrases, but in Esperanto each noun phrase can take its own la when it is definite.

Here de means of and shows possession or relationship.

So:

• la prezo de la nova komputilo = the price of the new computer

This is a very common way to link two nouns in Esperanto. Other examples:

• la koloro de la domo = the color of the house
• la nomo de la urbo = the name of the city

In Esperanto, adjectives usually come before the noun they describe.

So:

• nova komputilo = new computer

You can sometimes place the adjective after the noun, but the normal, neutral order is before it. A learner should generally put adjectives before nouns unless there is a special stylistic reason not to.

Because it is describing a singular noun in the basic form.

• komputilo ends in -o, so it is a singular noun
• nova matches it as an adjective, so it ends in -a

There is no -j because it is not plural, and no -n because it is not in the accusative.

Compare:

• la nova komputilo = the new computer
• la novaj komputiloj = the new computers
• mi aĉetas la novan komputilon = I am buying the new computer

gravas is the present-tense form of the verb gravi, which means to be important or to matter.

Breakdown:

• grav- = the root meaning important / weighty
• -i = infinitive, so gravi = to be important
• -as = present tense, so gravas = is important / matters

So La prezo ... gravas means The price ... matters or is important.

Both are possible, but they are built differently.

1. La prezo ... gravas por mi

   • literally: The price matters to me
   • here gravas is a verb

2. La prezo ... estas grava por mi

   • literally: The price is important to me
   • here grava is an adjective, and estas is the verb to be

Both are natural. The sentence you were given uses the verbal style: gravi.

Por mi means for me or, in smoother English here, to me.

It shows whose point of view is involved:

• gravas por mi = matters to me / is important to me

This is a common use of por. It does not mean physical direction here; it marks relevance or importance in relation to someone.

Normally, por mi is the better choice here.

• por mi = for me, from my perspective, in relation to me
• al mi usually means to me in a more directional sense

With gravi, Esperanto commonly uses por to express important to someone:

• Tio gravas por mi. = That matters to me.

Using al mi here would sound less standard to most learners and speakers.

Because there is no direct object here.

The sentence is built around the subject:

• La prezo de la nova komputilo = the subject

And the predicate:

• gravas por mi = matters to me / is important to me

Since nothing is being directly acted upon, no accusative is needed.

The verb is gravas, and the thing that matters is la prezo de la nova komputilo.

So that whole noun phrase is the subject:

• La prezo de la nova komputilo = the price of the new computer

If you turn it into a question, it becomes clearer:

• Kio gravas por mi?
• La prezo de la nova komputilo.

That shows that la prezo de la nova komputilo is the subject.

Yes. Komputilo is a normal Esperanto noun meaning computer.

It is built from:

• komput- = compute
• -il- = tool/instrument
• -o = noun ending

So literally it is something like a computing tool.

This is a very typical Esperanto word-building pattern. For example:

• tranĉi = to cut
• tranĉilo = a cutting tool, knife

Yes, Esperanto word order is fairly flexible, but the normal order here is the most natural:

• La prezo de la nova komputilo gravas por mi.

You might also hear:

• Por mi gravas la prezo de la nova komputilo.

That version puts more emphasis on por mi or makes the sentence flow differently in context. Because Esperanto has clearer endings than English, word order can vary more, but the neutral style is usually the simplest one.

In ordinary usage, prezo usually means price in the monetary sense.

So in this sentence, it most naturally refers to how much the new computer costs.

If you wanted to talk about broader value, Esperanto often uses valoro instead.

So:

• prezo = price
• valoro = value

That distinction is useful to remember.