Exercițiul acesta este greu, dar profesoara îl explică bine.

It is the definite article attached to the end of the noun. Romanian usually says the by adding it to the noun, not by putting a separate word in front.

• exercițiu = exercise / an exercise
• exercițiul = the exercise

So exercițiul acesta means this exercise.

In Romanian, this kind of demonstrative normally appears in one of two correct patterns:

• acest exercițiu = this exercise
• exercițiul acesta = this exercise

When the demonstrative comes after the noun, the noun is usually definite: exercițiul acesta.
When it comes before the noun, you normally use acest without the noun’s definite ending: acest exercițiu.

Acesta exercițiul is not the normal pattern.

Both mean this exercise. The difference is mostly one of style and emphasis.

• exercițiul acesta is very common and natural in everyday Romanian.
• acest exercițiu can sound a bit more formal, careful, or written.

In casual speech, you may also hear exercițiul ăsta, which is more colloquial.

Because Romanian neuter nouns behave like masculine in the singular and feminine in the plural.

So in the singular:

• exercițiul acesta
• este greu
• profesoara îl explică

All those singular forms are the ones also used with masculine nouns.

In the plural, the same noun would take feminine-type agreement:

• exercițiile acestea sunt grele
• profesoara le explică

So exercițiu is still neuter, but its agreement pattern changes depending on number.

The adjective has to agree with the noun. Here the noun is exercițiul, which is singular neuter, and singular neuter uses masculine-form agreement.

So:

• masculine singular / neuter singular: greu
• feminine singular: grea
• plural: grele

That is why Exercițiul acesta este greu is correct.

Because profesoara means the teacher (female), while profesoară is the basic form meaning teacher.

• o profesoară = a teacher
• profesoara = the teacher

Again, Romanian attaches the definite article to the end of the noun instead of using a separate word like the.

Îl is a direct object pronoun. Here it means it, referring back to exercițiul.

So:

• profesoara îl explică = the teacher explains it

This is the masculine singular direct-object clitic, which is also the form used for singular neuter nouns.

It can also mean him in other contexts, depending on what it refers to.

Because short unstressed object pronouns in Romanian usually go before the conjugated verb.

So Romanian says:

• îl explică
• o vede
• le spune

This is very normal word order. English puts the object after the verb (explains it), but Romanian often puts this short pronoun before the verb.

Because bine is an adverb, and it modifies the verb explică.

• explică bine = explains well

By contrast, bun is an adjective, so it describes a noun:

• un exercițiu bun = a good exercise
• o explicație bună = a good explanation

So in this sentence, bine is the correct form because it tells us how the teacher explains.

In standard Romanian, yes, it is necessary here.

• Exercițiul acesta este greu is correct.
• Leaving out este would not be standard Romanian in this kind of sentence.

Romanian usually keeps the present tense of a fi (to be) in normal statements like this.

Romanian word order is fairly flexible, but the sentence as given is the most neutral and natural version.

• Exercițiul acesta este greu, dar profesoara îl explică bine. = neutral
• Profesoara îl explică bine. = also fine as a standalone clause
• Dar îl explică bine profesoara. = possible, but it shifts emphasis

So yes, some movement is possible, but îl still usually stays before the verb, and the original sentence is a very natural default order.

They are important Romanian letters with their own sounds:

• ț = like ts in cats
  • exercițiul
• ă = a short neutral vowel, somewhat like the a in sofa
  • profesoara
• î = a vowel that English does not really have; it is a close central sound
  • îl

These diacritics matter. Without them, the spelling is incomplete or can become ambiguous, so it is best to learn and use them from the start.