He traído mi cuaderno a la biblioteca.

Here he does not mean he in English.

It is the first-person singular form of the auxiliary verb haber, so in this sentence it means I have as part of a compound tense:

• he = I have
• traído = brought

So he traído = I have brought

Important: this is haber, not tener.

• Tengo un cuaderno = I have a notebook
• He traído un cuaderno = I have brought a notebook

He traído is the present perfect:

• he
  • past participle (traído)

In Spanish from Spain, this tense is very commonly used for actions in the recent past or actions that still connect to the present.

So He traído mi cuaderno a la biblioteca suggests something like:

• I’ve brought my notebook to the library
• and that fact is relevant now

By contrast, traje is the preterite (simple past):

• Traje mi cuaderno a la biblioteca

That also means I brought my notebook to the library, but it sounds more like a completed past event, with less emphasis on its present relevance.

A useful Spain-vs-Latin-America note:

• In Spain, he traído is very natural in many everyday situations.
• In much of Latin America, speakers often prefer traje in the same context.

Spanish usually distinguishes between bringing and taking more clearly:

• traer = to bring, to carry something toward the speaker or the reference point
• llevar = to take, to carry something away from the speaker or reference point

So:

• He traído mi cuaderno a la biblioteca = I have brought my notebook to the library
• He llevado mi cuaderno a la biblioteca = I have taken my notebook to the library

In practice, traer makes sense if the speaker is at the library now, or is treating the library as the relevant destination from the current point of view.

Because a marks movement toward a destination.

• a la biblioteca = to the library
• en la biblioteca = in/at the library

So:

• He traído mi cuaderno a la biblioteca = I brought my notebook to the library
• Estoy en la biblioteca = I am at/in the library

English sometimes uses to and at very naturally without much thought, but in Spanish this distinction is important:

• a = destination
• en = location

Because Spanish often omits subject pronouns when they are not needed.

The verb form he already tells you the subject is yo:

• he = I have

So He traído mi cuaderno a la biblioteca already clearly means I have brought my notebook to the library.

You could say:

• Yo he traído mi cuaderno a la biblioteca

but that usually adds emphasis, contrast, or clarification, for example:

• Yo he traído mi cuaderno, pero él no.

Because mi is the correct possessive adjective before a noun.

Use:

• mi cuaderno = my notebook

But mío is usually used:

1. on its own
   • El cuaderno es mío = The notebook is mine
2. after the noun for emphasis or style
   • un cuaderno mío = a notebook of mine

So before the noun, the normal form is:

• mi cuaderno

not:

• mío cuaderno

The accent mark shows the correct stress and pronunciation.

Traído is pronounced in three syllables:

• tra-í-do

The written accent on í shows that aí is not pronounced as one diphthong. Instead, the í forms its own syllable.

This is the same pattern as in:

• caído
• leído
• oído

Without the accent, a learner might try to pronounce it too quickly as one combined vowel sound.

Yes, normally it is.

Biblioteca is a common noun, so Spanish usually uses the definite article here:

• a la biblioteca = to the library

Leaving out the article would usually sound wrong in a normal sentence:

• He traído mi cuaderno a biblioteca ❌

This is different from some place names or special fixed expressions, but with ordinary nouns like biblioteca, escuela, casa, oficina, the article is often required depending on the structure.

Cuaderno usually means notebook.

It often refers to:

• a school notebook
• an exercise book
• a notebook for writing notes

It is not the same as:

• libro = book
• carpeta = folder/binder
• agenda = planner/diary

So mi cuaderno is most naturally my notebook.

Yes, but the original order is the most neutral and natural:

• He traído mi cuaderno a la biblioteca.

This is basically:

• verb + object + destination

Spanish word order is more flexible than English, so other versions are possible, but they may sound more marked or emphasize something different, for example:

• A la biblioteca he traído mi cuaderno.
  More emphasis on to the library

• Mi cuaderno lo he traído a la biblioteca.
  More emphasis on my notebook

For a learner, the original version is the safest default.

A simple guide is:

• He → the h is silent
• traído → pronounced roughly tra-EE-do
• cuaderno → roughly kwa-DER-no
• biblioteca → roughly bee-blee-o-TE-ka

A fuller approximation might be:

• eh tra-EE-do mee kwa-DER-no a la bee-blee-o-TE-ka

A few useful pronunciation points:

• h is silent in Spanish
• tr is pronounced together, not like English ch
• r in traído and cuaderno is a tapped Spanish r
• qu in cuaderno sounds like kw because of the u
• in biblioteca, the c sounds like k because it comes before a