Ĉi tiu meblo estas malnova sed utila.

In Esperanto, this is usually formed with tiu (that / that one) plus ĉi, which adds the sense of here / near the speaker.

So:

• tiu = that
• ĉi tiu or tiu ĉi = this

Both ĉi tiu meblo and tiu ĉi meblo are correct.
Ĉi is not normally used by itself as a determiner before a noun.

Esperanto does have a definite article: la. But you do not use la together with ĉi tiu here, because ĉi tiu already makes the noun definite.

So:

• la meblo = the piece of furniture
• ĉi tiu meblo = this piece of furniture

Using both would be unnecessary in normal Esperanto.

Meblo means a piece of furniture or an item of furniture.

The ending -o is the normal noun ending in Esperanto.
That means meblo is a noun.

Examples:

• libro = book
• domo = house
• meblo = piece of furniture

If it were plural, it would become mebloj.

Estas is the present tense form of esti, which means to be.

Esperanto verbs do not change according to the subject, so the same form is used with I, you, he, she, it, we, they.

Examples:

• mi estas = I am
• vi estas = you are
• ĝi estas = it is
• ĉi tiu meblo estas = this piece of furniture is

That makes Esperanto verb conjugation much simpler than English.

The ending -a marks an adjective in Esperanto.

So:

• malnova = old
• utila = useful

Adjectives in Esperanto usually agree with the noun they describe in number and case. Since meblo is singular and has no accusative -n, the adjectives are also just:

• malnova
• utila

If the noun were plural, you would say:

• Ĉi tiuj mebloj estas malnovaj sed utilaj.

Malnova is built from:

• nova = new
• mal- = the direct opposite

So:

• nova = new
• malnova = old

Mal- does not just mean simple not. It usually creates a strong opposite.

More examples:

• bona = good → malbona = bad
• granda = big → malgranda = small
• varma = warm → malvarma = cold

This is a very common and useful pattern in Esperanto.

Sed means but. It connects two ideas that contrast with each other:

• the furniture is old
• but it is useful

So sed shows contrast.
If you used kaj, it would just mean and, with no contrast:

• malnova kaj utila = old and useful

Both are possible in different contexts, but sed emphasizes the contrast.

Because there is no direct object here.

The sentence has:

• a subject: Ĉi tiu meblo
• a verb: estas
• predicate adjectives: malnova and utila

After esti (to be), Esperanto uses predicate nouns/adjectives, not direct objects. So no accusative -n is needed.

Compare:

• Mi vidas ĉi tiun meblon. = I see this piece of furniture.
  Here you do get -n, because meblon is the direct object.

But in your sentence, the furniture is simply being described, not acted upon.

A few key points:

• ĉ sounds like ch in church
• tiu is roughly TEE-oo
• stress in Esperanto normally falls on the second-to-last syllable

A rough pronunciation guide:

• Ĉi = chee
• tiu = TEE-oo
• meblo = MEB-lo
• estas = ES-tas
• malnova = mal-NO-va
• sed = sed
• utila = u-TEE-la

So the sentence sounds roughly like:

chee TEE-oo MEB-lo ES-tas mal-NO-va sed u-TEE-la

Yes. Ĉi tiu and tiu ĉi are both correct and mean the same thing: this.

So both of these are fine:

• Ĉi tiu meblo estas malnova sed utila.
• Tiu ĉi meblo estas malnova sed utila.

In modern usage, many learners first meet ĉi tiu, but tiu ĉi is also standard Esperanto.

Because it describes the noun through the verb estas.

In English, after to be, we use adjectives:

• The furniture is old
• The furniture is useful

Esperanto works the same way:

• estas malnova
• estas utila

An adverb would end in -e and would describe how something is done, not what something is like.

For example:

• utile = usefully / in a useful way

But here you need the adjective utila, because you are describing the furniture itself.