Ni vivas en ĉi tiu jarcento.

In Esperanto, -i is the infinitive ending, like English to live.
So vivi means to live in a general dictionary form.

But in a normal sentence, you usually need a finite verb. The ending -as marks the present tense, so:

• vivi = to live
• vivas = live / are living

So Ni vivas means we live or we are living.

Here is the breakdown:

• Ni = we
• vivas = live / are living
• en = in
• ĉi tiu = this
• jarcento = century

So the sentence is built very directly:

• Ni → subject
• vivas → verb
• en ĉi tiu jarcento → prepositional phrase meaning in this century

En is the preposition for in.

So:

• en la domo = in the house
• en la urbo = in the city
• en ĉi tiu jarcento = in this century

It is used here because the sentence places us inside a time period, just as en can place something inside a physical space.

In English, we also say in this century, so this part matches English quite well.

Ĉi tiu is the normal way to say this.

It is made of:

• tiu = that
• ĉi = a particle that brings it close, giving the sense of this

So:

• tiu = that one
• ĉi tiu = this one

Esperanto often uses ĉi with demonstratives to show nearness.

You may also see the reversed form:

• tiu ĉi

That means the same thing. Both are correct, but ĉi tiu is very common and often feels more natural to learners.

There is no -n because jarcento is the object of the preposition en, not a direct object.

In Esperanto:

• Direct objects usually take -n
• Nouns after prepositions usually do not take -n

So:

• Mi vidas la domon. = I see the house.
  Here domon gets -n because it is a direct object.

But:

• Mi estas en la domo. = I am in the house.
  Here domo does not take -n because it follows en.

Therefore:

• en ĉi tiu jarcento is correct, not en ĉi tiun jarcenton

Because ĉi tiu already makes the noun definite.

Compare:

• jarcento = a century / century
• la jarcento = the century
• ĉi tiu jarcento = this century

In Esperanto, you normally do not combine la with ĉi tiu in this kind of phrase.
So ĉi tiu jarcento already means a specific century: this century.

Jarcento is built from:

• jar- = year
• cent = hundred
• -o = noun ending

So it literally means something like a hundred-year unit, which is why it means century.

This kind of word-building is very common in Esperanto and is worth noticing.

Related words:

• jaro = year
• cento = a hundred
• jarcento = century

Yes. Esperanto word order is fairly flexible, because the grammar endings make the structure clear.

The neutral order here is:

• Ni vivas en ĉi tiu jarcento.

But you could also say:

• En ĉi tiu jarcento ni vivas.

That puts more emphasis on en ĉi tiu jarcento.

Both are grammatical. The original version is just the most straightforward, neutral way to say it.

A rough guide for an English speaker:

• ĉ sounds like ch in church
• j sounds like y in yes
• c sounds like ts in cats
• Esperanto stress is usually on the second-to-last syllable

So:

• ĉi sounds roughly like chee
• tiu sounds roughly like TEE-oo
• jarcento sounds roughly like yar-TSEN-to, with stress on TSEN

And the whole sentence is roughly:

• Nee VEE-vas en chee TEE-oo yar-TSEN-to

That is only an approximation, but it is a useful starting point.

It can mean either, depending on context.

Esperanto -as gives the present tense, but it does not force the same distinction English often makes between:

• we live
• we are living

So Ni vivas can be understood as:

• We live
• We are living

The surrounding context tells you which English translation sounds best. In this sentence, We live in this century is the most natural translation.