Discipuli disputationi de bono communi libenter intersunt.

Discipuli is nominative plural, so it is the subject of the sentence.

• discipulus = student / pupil
• discipuli = students

The verb intersunt is also plural, so it matches discipuli.

Because intersum takes the dative.

The verb intersum, interesse, interfui means to be present at, to take part in, or to participate in. Unlike English attend or join, it does not take a direct object in the accusative. Instead, the thing or event someone is present at is put in the dative.

So:

• disputatio = discussion / debate
• disputationi = to/for the discussion → idiomatically, at the discussion or in the discussion

Intersunt is the 3rd person plural present active indicative of intersum.

It breaks down like this:

• inter- = among, between
• sum = I am

So intersunt literally means something like they are among / they are present, and in normal English it is often translated as:

• they are present at
• they attend
• they take part in

Because the subject is discipuli (students), intersunt means the students take part / are present.

Because intersum is a compound verb in Latin.

Latin often combines a preposition or prefix with a verb:

• sum = I am
• adsum = I am present
• absum = I am absent
• intersum = I am in the midst of / I take part in / I am present at

So here intersunt is not inter plus a separate verb in the sentence; it is the normal verb form of intersum.

De takes the ablative, so both words are ablative singular:

• bono = ablative singular of bonum / bonus
• communi = ablative singular of communis

So de bono communi means about the common good.

This is a very common Latin pattern:

• de amico = about the friend
• de bello = about the war
• de virtute = about virtue

Yes. It is an adjective-noun phrase in the ablative singular after de.

• bonum can mean good as a noun
• commune means common or shared

Together, bonum commune means the common good, a very important political and philosophical expression in Latin.

In this sentence it appears as de bono communi because de requires the ablative.

Libenter is an adverb meaning:

• gladly
• willingly
• with pleasure

It modifies the verb intersunt.

Latin word order is more flexible than English word order, so libenter can often be placed in different positions without changing the basic meaning. Here it sits between the phrase about the discussion and the verb, which is perfectly normal Latin style.

Because classical Latin has no definite article and no indefinite article.

So Latin does not normally have separate words for:

• the
• a
• an

That means:

• discipuli can mean students or the students
• disputatio can mean discussion or the discussion

The context or the translation tells you which is most natural in English.

Yes. Because Latin and English do not line up word-for-word, several translations can be correct. For example:

• The students gladly take part in the discussion about the common good.
• The students willingly participate in the debate about the common good.
• The students are gladly present at the discussion about the common good.

The most natural English version usually uses take part in or participate in.

The word order is meaningful stylistically, but the grammar is mainly shown by the endings.

The sentence is:

• Discipuli — subject
• disputationi — dative with intersunt
• de bono communi — prepositional phrase
• libenter — adverb
• intersunt — verb

A more literal order might be:

The students, in the discussion about the common good, gladly take part.

Latin often places the verb at or near the end, and that is what happens here.