Necesito un bolígrafo para rellenar el formulario.

In Spanish, the subject pronoun is often omitted because the verb ending already tells you who the subject is.

• Necesito = I need
• The -o ending shows it is yo

So Necesito un bolígrafo is completely natural. You can say Yo necesito if you want extra emphasis, for example:

• Yo necesito un bolígrafo, no un lápiz.
  I need a pen, not a pencil.

Un means a or one and is the masculine singular indefinite article.

• un bolígrafo = a pen

It is un because bolígrafo is a masculine singular noun. If the noun were feminine, you would normally use una.

For example:

• un bolígrafo
• una carpeta

You usually learn the gender together with the noun: el bolígrafo.

Many Spanish nouns ending in -o are masculine, and bolígrafo follows that common pattern:

• el bolígrafo
• un bolígrafo

A good habit is to memorise nouns with their article:

• not just bolígrafo
• but el bolígrafo

That makes it easier to use them correctly in sentences.

The accent mark shows where the stress goes in pronunciation.

• bolígrafo is stressed on lí
• It sounds roughly like boh-LEE-grah-fo

Without the accent mark, Spanish stress rules would suggest a different pronunciation, so the written accent is necessary.

Para often means for or in order to, and when it is followed by an infinitive, it expresses purpose.

So:

• para rellenar = to fill in / in order to fill in

This structure is very common:

• Necesito agua para beber. = I need water to drink.
• Tengo dinero para comprarlo. = I have money to buy it.

After para to express purpose, Spanish uses the infinitive.

So:

• para rellenar
• not a conjugated form like para relleno or para rellena

This is similar to English to fill in.

The pattern is:

• para + infinitive

Examples:

• para estudiar
• para comer
• para escribir

All three can be possible, but they are not always identical in tone or usage.

In Spain, with forms, these are common:

• rellenar un formulario = to fill in a form
• llenar un formulario = also understandable, but rellenar is especially common for forms
• completar un formulario = to complete a form

A rough difference:

• rellenar focuses on writing in the blank spaces
• completar focuses on finishing it

So rellenar el formulario is a very natural choice.

Formulario is a masculine noun, so it takes:

• el = the
• un = a

So:

• el formulario = the form
• un formulario = a form

Again, the -o ending is a strong clue that the noun is masculine.

That depends on context.

• un formulario would mean a form
• el formulario means the form

In this sentence, el formulario suggests a specific form the speaker has in mind, probably one already known from the situation.

For example:

• Necesito un bolígrafo para rellenar el formulario.
  I need a pen to fill in the form.

If no specific form were intended, un formulario could also be possible in another context.

In Spain, bolígrafo is the standard word for pen.

In some other Spanish-speaking countries, pluma or lapicero may be used more often. But for Spanish from Spain, bolígrafo is the safest and most natural choice.

You may also hear the informal short form:

• boli = pen

So in casual speech, someone might say:

• Necesito un boli.

No, formulario is the normal word for form in this context.

• formulario = form, document with spaces to fill in

Using forma here would usually sound wrong, because forma normally means shape, form, or manner, not an administrative document.

So for paperwork, use:

• formulario

It sounds very natural and correct in Spanish from Spain.

Necesito un bolígrafo para rellenar el formulario is exactly the kind of sentence a native speaker might say in an office, reception area, bank, school, or government building.

A slightly more informal version could be:

• Necesito un boli para rellenar el formulario.

But the original sentence is perfectly normal and appropriate.