Отправитель письма забыл прикрепить вложение, и получатель ничего не понял.

Because отправитель is being specified by another noun: the sender of the letter/email. In Russian, this of relationship is usually shown with the genitive case.

• письмо = letter, email
• письма = of the letter/email

So отправитель письма literally means sender of the letter/email.

The same pattern appears in:

• автор книги = author of the book
• получатель посылки = recipient of the package

It can mean either letter or email, depending on context.

In this sentence, вложение strongly suggests email, because attachments are normally associated with emails. So although письмо by itself is ambiguous, the whole sentence is most naturally understood as being about an email.

If Russian wants to be extra explicit, it can say:

• электронное письмо = email

Yes. Вложение is a standard word for an attachment, especially in email context.

A learner should be careful not to confuse it with приложение, which often means:

• app/application
• appendix/addendum

So for an attached file in an email, вложение is the safest word.

This is a very common pattern:

забыть + infinitive = to forget to do something

So:

• забыл прикрепить вложение = forgot to attach the attachment

That is different from:

• забыл о вложении = forgot about the attachment

And also different from:

• забыл вложение = forgot the attachment itself / left it behind

So the infinitive прикрепить is important because it shows the forgotten action.

Because the action is understood as one complete act: attaching the file.

• прикрепить = perfective, to attach successfully / as a complete action
• прикреплять = imperfective, to be attaching / to attach repeatedly or habitually

Here the sender forgot to do one specific thing once, so прикрепить is the natural choice.

A useful comparison:

• Он забыл прикрепить файл. = He forgot to attach the file.
• Он часто забывал прикреплять файлы. = He often forgot to attach files.

It is the direct object of прикрепить, so it is in the accusative case. But вложение is a neuter inanimate singular noun, and for that type of noun, the nominative and accusative forms are identical.

So:

• nominative: вложение
• accusative: вложение

The case is different grammatically, but the spelling stays the same.

Because ничто is the nominative form, while here the word is not the subject. It is the object of the verb не понял.

In this kind of negative sentence, Russian normally uses ничего:

• ничего не понимаю
• ничего не вижу
• ничего не понял

So ничего не понял is the standard pattern. A learner will hear it very often.

Because Russian normally uses negative agreement: if there is a negative pronoun like ничего, the verb also takes не.

So standard Russian says:

• ничего не понял
• никого не видел
• никогда не был там

This is different from standard English, where double negatives are usually avoided. In Russian, this is not a mistake; it is the normal grammar.

Because и is joining two separate clauses, each with its own subject and verb:

• Отправитель письма забыл прикрепить вложение
• получатель ничего не понял

Since these are two independent parts of the sentence, Russian puts a comma before и.

If и only connected two verbs with the same subject, there would usually be no comma.

Russian past tense verbs agree with the subject in gender and number.

Here the subjects are:

• отправитель — masculine singular
• получатель — masculine singular

So the verbs are:

• забыл
• понял

Compare:

• отправительница забыла = the female sender forgot
• получательница поняла = the female recipient understood
• отправители забыли = the senders forgot

Because Russian has no articles. It does not have separate words for a/an and the.

Whether something is specific or general is usually understood from:

• context
• word order
• the situation itself

So отправитель, получатель, and вложение can all be translated with the in English, even though Russian does not say it explicitly.

Yes. Russian word order is relatively flexible because case endings show the grammatical relationships.

The original order is neutral and natural:

• Отправитель письма забыл прикрепить вложение, и получатель ничего не понял.

But other orders are possible for emphasis. For example:

• Получатель ничего не понял, потому что отправитель письма забыл прикрепить вложение.
• Ничего не понял получатель... — this puts extra focus on the recipient

So the given sentence is not the only possible order; it is just a straightforward, neutral one.