Ég hendi gömlum pappírum, en hún vill ekki henda gömlu póstkortunum.

They are two forms of the same verb, henda.

• hendi = I throw / I toss → 1st person singular, present tense
• henda = to throw / to toss → infinitive

So:

• Ég hendi gömlum pappírum = I throw old papers
• hún vill ekki henda ... = she does not want to throw ...

This is similar to English I throw vs want to throw.

Because henda takes a dative object in Icelandic.

That means the thing being thrown is put in the dative case:

• pappírum = dative plural of pappír
• póstkortunum = dative plural definite of póstkort

So after henda, you do not use the nominative form of the noun.

A learner-friendly way to remember it is:

• henda einhverju = throw something
  where einhverju is dative

Both adjective forms are plural and dative, but they belong to different adjective patterns:

• gömlum = strong adjective form
• gömlu = weak adjective form

Why the difference?

1. gömlum pappírum is indefinite: old papers
2. gömlu póstkortunum is definite: the old postcards

In Icelandic, adjectives usually:

• use the strong form with an indefinite noun
• use the weak form with a definite noun

So:

• gömlum pappírum = old papers
• gömlu póstkortunum = the old postcards

In Icelandic, the is often added to the end of the noun as a suffix rather than written as a separate word.

So:

• póstkort = postcard / postcards
• póstkortum = postcards, dative plural
• póstkortunum = the postcards, dative plural

The ending -unum tells you this is a definite form, roughly corresponding to English the.

This is very common in Icelandic:

• hús = house
• húsið = the house

Because vilja is a modal-type verb, and after it Icelandic normally uses the bare infinitive, without að.

So you say:

• hún vill henda = she wants to throw

not:

• hún vill að henda

This is similar to English:

• she wants to throw not
• she wants to to throw

Compare with other Icelandic verbs that do require að before the infinitive, such as:

• reyna að henda = try to throw

Because ekki usually comes after the finite verb and before the infinitive or other material it affects.

Here the finite verb is vill:

• hún vill ekki henda ...

So the structure is:

• hún = subject
• vill = finite verb
• ekki = not
• henda = infinitive

This is a very common Icelandic pattern.

A useful comparison:

• Hún kemur ekki. = She is not coming.
• Hún vill ekki koma. = She does not want to come.

Because vilja is the infinitive, while vill is the conjugated form used with hún.

The verb is irregular:

• ég vil = I want
• þú vilt = you want
• hann/hún/það vill = he/she/it wants

So in the sentence:

• hún vill ekki henda ... = she does not want to throw ...

You must use vill, because the subject is hún.

En means but.

It connects two main clauses and shows contrast:

• Ég hendi gömlum pappírum = I throw old papers
• en hún vill ekki henda gömlu póstkortunum = but she does not want to throw the old postcards

So en is a coordinating conjunction, like English but.

The basic forms are:

• pappír = paper, masculine
• póstkort = postcard, neuter

Their relevant forms here are:

pappír

• singular nominative: pappír
• plural nominative: pappírar
• plural dative: pappírum

póstkort

• singular nominative: póstkort
• plural nominative: póstkort
• plural dative: póstkortum
• plural dative definite: póstkortunum

Knowing the noun’s gender helps you predict many forms, although in this sentence the adjective difference is mainly about strong vs weak, not masculine vs neuter.

It can mean any of those, depending on context.

In everyday usage, henda often means:

• throw
• toss
• throw away
• discard

So in this sentence, if the shown meaning is about getting rid of things, throw away or discard is a natural English translation.

The important grammar point is that henda still takes the object in the dative case.

Because this is weak dative plural, and in that form the adjective ending is -u for all genders.

So even though póstkort is neuter, the adjective is still:

• gömlu

not something specially neuter-looking.

Likewise, in gömlum pappírum, the ending -um is the strong dative plural form, also shared across genders.

So in this sentence, the key difference is not masculine vs neuter, but:

• strong adjective: gömlum
• weak adjective: gömlu