Ni uzas modernan aparaton.

Because aparaton is the direct object of the verb, and Esperanto marks direct objects with -n.
The adjective modernan describes aparaton, so it must match the noun in both number and case.

So:

• aparato = device
• aparaton = device as a direct object
• moderna aparato = a modern device
• modernan aparaton = a modern device as a direct object

This matching of adjective and noun is a very important Esperanto rule.

Ni means we.

In Esperanto, verb endings do not change depending on the subject, so uzas could mean I use, you use, we use, they use, and so on unless the subject is stated.

That is why ni is usually included here: it makes clear that the subject is we.

Esperanto verbs do not change according to person or number.

So:

• mi uzas = I use
• vi uzas = you use
• li uzas = he uses
• ni uzas = we use
• ili uzas = they use

The ending -as always marks the present tense.
That makes Esperanto much simpler than English in this area.

The sentence divides like this:

• Ni = subject
• uzas = verb
• modernan aparaton = direct object

So the structure is:

subject + verb + object

That is also a very common word order in Esperanto, just as in English.

Esperanto does have the definite article la, which means the, but it has no indefinite article like English a/an.

So:

• aparato can mean a device or just device, depending on context
• la aparato means the device

In this sentence, there is no la, so it is understood as a modern device or simply modern device, depending on the translation context.

Moderna is an adjective, meaning modern.

Esperanto adjectives normally end in -a:

• moderna = modern
• granda = big
• nova = new

Adjectives must agree with the noun they describe:

• in number: singular or plural
• in case: especially whether the noun has -n

So:

• moderna aparato = a modern device
• modernaj aparatoj = modern devices
• modernan aparaton = a modern device as object
• modernajn aparatojn = modern devices as objects

Yes. Esperanto allows fairly flexible adjective placement.

So both of these are grammatical:

• modernan aparaton
• aparaton modernan

However, the first one is much more natural and common in ordinary speech and writing.
Learners should usually put the adjective before the noun unless there is a special stylistic reason not to.

In Esperanto, plural nouns end in -j.

So:

• aparato = device
• aparatoj = devices

Since the sentence has aparaton and not aparatojn, it is:

• singular: one device
• direct object: marked by -n

If it were plural, it would be modernajn aparatojn.

Yes, it is very close in meaning.
The Esperanto verb is uzi = to use.

Its forms include:

• uzi = to use
• uzas = use / uses / are using
• uzis = used
• uzos = will use
• uzu = use! / should use
• uzus = would use

This regular pattern is one of the easiest parts of Esperanto grammar.

A simple pronunciation guide is:

Nee OO-zahs moh-der-NAHN ah-pah-RAH-tohn

A few helpful points:

• ni sounds like nee
• u is like oo in food
• z is always a voiced z
• a is like a in father
• stress in Esperanto usually falls on the second-to-last syllable

So:

• uzas → U-zas
• modernan → mo-der-NAN
• aparaton → a-pa-RA-ton

Normally, no. The -n is a real grammatical marker, not something optional.

Without it, you would get:

• Ni uzas moderna aparato

That is ungrammatical, because the direct object should be marked, and the adjective must match the noun.

The correct form is:

• Ni uzas modernan aparaton

So even if the meaning seems obvious, Esperanto grammar still requires the endings.