Doctrina bona discipulis utilis est.

Here is the grammatical role of each word:

• doctrina — the subject; nominative singular feminine
• bona — an adjective modifying doctrina; nominative singular feminine
• discipulis — dative plural; for the students or to the students
• utilis — a predicate adjective with est; it describes doctrina
• est — is

So the basic structure is:

[subject] + [adjective modifying subject] + [dative] + [predicate adjective] + [verb]

In very plain grammatical English: Good teaching is useful to the students.

Because bona agrees with doctrina.

Doctrina is:

• feminine
• singular
• nominative

So the adjective modifying it must also be:

• feminine
• singular
• nominative

The adjective is bonus, bona, bonum = good.
Its feminine nominative singular form is bona.

So:

• doctrina bona = good teaching / good instruction

Because utilis belongs to a different adjective pattern from bonus, bona, bonum.

It is a third-declension adjective:

• utilis for masculine/feminine
• utile for neuter

Since doctrina is feminine singular, the correct form is utilis.

Even though bona and utilis both agree with doctrina, they do not have to look alike, because they come from different adjective types.

So:

• bona = feminine nominative singular of a first/second-declension adjective
• utilis = feminine nominative singular of a third-declension adjective

Because of case and sentence structure.

Utilis is nominative singular feminine, which matches doctrina.
Discipulis is dative plural, so it does not match utilis.

Also, with est, Latin often uses a predicate adjective describing the subject:

• doctrina ... utilis est = teaching is useful

So utilis is saying something about doctrina, while discipulis tells us useful to whom.

Because utilis commonly takes the dative in Latin.

Latin often expresses useful to/for someone with:

• utilis + dative

So:

• discipulis utilis = useful to the students / useful for the students

An English speaker often expects a separate word like to or for, but Latin can show that idea just by case ending.

Here discipulis is the dative plural of discipulus = student.

Because it is not nominative.

In Latin, the subject is usually in the nominative case.
Here:

• doctrina = nominative singular
• discipulis = dative plural

So doctrina is the subject, not discipulis.

This also explains why the verb is est (is) and not a plural verb:

• the subject doctrina is singular

Yes, it could absolutely be bona doctrina.

Latin adjective position is much more flexible than English word order. Both of these are possible:

• doctrina bona
• bona doctrina

Both mean good teaching.

Sometimes word order can slightly affect emphasis or style, but not the basic grammar here. A learner should mainly focus on the endings:

• doctrina and bona match, so they belong together

Because Latin relies much more on endings than on fixed word order.

English needs word order to show meaning:

• Good teaching is useful for students

Latin can move things around more freely because the forms already show their jobs:

• doctrina = subject
• discipulis = dative
• utilis = predicate adjective
• est = verb

So Doctrina bona discipulis utilis est is natural Latin, but other orders are also possible, such as:

• Bona doctrina discipulis utilis est
• Discipulis doctrina bona utilis est

The exact emphasis may shift, but the core meaning stays the same.

Because classical Latin does not have articles like English the and a/an.

So doctrina can mean:

• teaching
• the teaching
• a teaching
• instruction
• doctrine

The context tells you which English wording is best.

The same is true for discipulis:

• to students
• to the students

English usually forces you to choose an article; Latin often does not.

Sometimes, yes. Latin can omit a form of to be when the meaning is obvious, especially in certain styles.

So a shortened version like:

Doctrina bona discipulis utilis

could still be understood as: Good teaching is useful to students.

However, for a straightforward textbook sentence, including est is completely normal and often clearer for beginners.

Doctrina can have several related meanings, depending on context, such as:

• teaching
• instruction
• learning
• doctrine

In a sentence like this, the most natural English choice is often teaching or instruction.

That kind of range is normal in Latin vocabulary: one Latin word may correspond to several English words, and context decides which fits best.