Puer se timidum vocari non vult.

Here is the sentence broken down:

• Puer = the boy; nominative singular, the subject of vult
• se = himself; reflexive pronoun, accusative
• timidum = timid / shy; accusative singular masculine
• vocari = to be called; present passive infinitive of vocare
• non vult = does not want

So the structure is:

• main clause: Puer ... non vult = The boy does not want
• infinitive clause: se timidum vocari = himself to be called timid

That infinitive clause is the thing he does not want.

Se is the reflexive pronoun, meaning himself / herself / itself / themselves, depending on context.

In this sentence, it refers back to puer, the subject of the main verb:

• Puer ... se ... vult = The boy ... wants himself ...

Latin uses se when the person affected is the same as the subject of the main clause.

Compare:

• Puer se timidum vocari non vult = The boy does not want himself to be called timid.
• Puer eum timidum vocari non vult = The boy does not want him to be called timid.
  That would mean some other male person, not the boy himself.

Because timidum goes with se, and se is accusative.

With verbs like vocare meaning to call or to name, Latin often uses a predicate accusative: the word describing the person called is put in the same case as that person.

For example:

• aliquem timidum vocare = to call someone timid

Here, inside the infinitive phrase:

• se = accusative
• timidum = accusative, agreeing with se

So se timidum vocari means himself to be called timid.

Vocari is the present passive infinitive of vocare.

• vocare = to call
• vocari = to be called

So:

• timidum vocare = to call someone timid
• timidum vocari = to be called timid

The ending -ari is the regular present passive infinitive ending for a first-conjugation verb like vocare.

No. Because vocari is passive, not active.

• vocare = to call
• vocari = to be called

So:

• Puer se timidum vocare non vult would mean The boy does not want to call himself timid
• Puer se timidum vocari non vult means The boy does not want to be called timid

That one extra i matters.

Yes, it uses the same basic pattern.

Latin often uses an infinitive clause after verbs such as want, say, think, know, and similar verbs. When that infinitive has its own stated subject, that subject is often put in the accusative.

Here:

• se is the subject of vocari
• timidum is the predicate word linked with se
• vocari is the infinitive

So se timidum vocari is an infinitive clause meaning himself to be called timid.

English usually does not show this pattern so clearly, so it can feel unusual at first.

Because non is negating the main verb vult:

• non vult = does not want

That is the central idea of the sentence: the boy does not want something.

Latin word order is flexible, but placing non before the finite verb is very normal and clear here.

Because Latin relies much more on case endings and verb forms than on word order.

English depends heavily on position:

• The boy does not want to be called timid

Latin can move words more freely because their endings show their jobs:

• Puer is the subject
• se and timidum are accusative
• vocari is an infinitive
• vult is the main finite verb

So Latin can say:

• Puer se timidum vocari non vult
• Puer non vult se timidum vocari
• Se timidum vocari puer non vult

These can all mean basically the same thing, though emphasis may shift.

Because Latin infinitives are single words.

In English, we say:

• to call
• to be called

In Latin, those are built into the verb form itself:

• vocare = to call
• vocari = to be called

So Latin does not need a separate word meaning to here.

Because it agrees with se, and se refers back to puer, which is masculine singular.

So the form must match:

• puer → se timidum
• puella → se timidam
• pueri → se timidos
• puellae → se timidas

Even though se itself does not show gender, the adjective does.

A very literal unpacking is:

• Puer = the boy
• se = himself
• timidum = timid
• vocari = to be called
• non vult = does not want

So, word-for-word:

The boy does not want himself to be called timid.

That is a good bridge to the smoother English meaning.