После ужина я отдам тебе этот подарок.

Because после requires the genitive case.
The noun ужин means dinner, and its genitive singular form is ужина.

So:

• ужин = dinner
• после ужина = after dinner

This is a very common pattern in Russian:

• после работы = after work
• после школы = after school
• после фильма = after the movie

Тебе is the dative case of ты.

It is used because the verb отдать means to give / hand over / give back, and the person who receives something is normally in the dative case.

So in this sentence:

• я = I
• отдам = will give
• тебе = to you
• этот подарок = this gift

A useful comparison:

• Я дам тебе книгу. = I will give you a book.
• Я отдам тебе книгу. = I will give/hand the book to you, often with a sense of passing it over or returning it

Этот подарок is in the accusative case, because it is the direct object of the verb отдам.

In this sentence, the thing being given is this gift.

For masculine inanimate nouns like подарок, the accusative looks the same as the nominative:

• nominative: этот подарок
• accusative: этот подарок

So even though the form does not change here, the grammatical role is still accusative.

It is a future form because отдать is a perfective verb.

In Russian:

• imperfective verbs form the future with быть
  • infinitive
• perfective verbs form the simple future directly

So:

• отдавать = imperfective, to be giving / to give repeatedly
• отдать = perfective, to give / hand over once, as a completed action

The form отдам means I will give / I’ll hand over.

Compare:

• Я буду отдавать... = I will be giving / I will give repeatedly
• Я отдам... = I will give (one completed action)

Both can mean to give, but отдать often has an extra sense of:

• handing something over
• giving something away
• returning something to someone
• transferring possession more clearly

So:

• дать = give
• отдать = give over, hand over, give back, return

In many contexts, both may be possible, but отдать can sound more like the object is being passed into someone else’s hands.

For example:

• Я дам тебе подарок. = I’ll give you a gift.
• Я отдам тебе подарок. = I’ll hand the gift over to you.

Yes, Russian often can drop subject pronouns when they are clear from the verb form.

So both of these are possible:

• После ужина я отдам тебе этот подарок.
• После ужина отдам тебе этот подарок.

Because отдам already clearly means I will give, the pronoun я is not always necessary.

Including я can:

• add clarity
• add emphasis
• make the sentence feel a little more explicit

Yes. Russian word order is much more flexible than English word order because cases show the grammatical roles.

The neutral version here is:

• После ужина я отдам тебе этот подарок.

But other orders are possible, depending on emphasis:

• Я отдам тебе этот подарок после ужина.
• Этот подарок я отдам тебе после ужина.
• Тебе я отдам этот подарок после ужина.

These all mean roughly the same thing, but the focus changes.

For example:

• Тебе first can emphasize to you
• Этот подарок first can emphasize this gift
• После ужина first sets the time frame right away

Putting после ужина first is a natural way to set the scene or time at the start.

Russian often places time expressions early in the sentence, especially when they provide background information.

So this structure feels very natural:

• После ужина = after dinner
• then the main statement follows

English does this too sometimes:

• After dinner, I’ll give you this gift.

Yes, этот means this, while тот means that.

Here:

• этот подарок = this gift
• тот подарок = that gift

Like adjectives, этот changes to match the noun in gender, number, and case.

For example:

• этот подарок = this gift
• эта книга = this book
• это письмо = this letter
• эти вещи = these things

A noun ending in a consonant is usually masculine in Russian.

Since подарок ends in -к (a consonant sound), it is masculine.

That is why the demonstrative is этот, not эта or это.

Compare:

• этот подарок = this gift, masculine
• эта ручка = this pen, feminine
• это окно = this window, neuter

Yes, stress is important in Russian. A helpful stress guide is:

• послЕ
• Ужина
• я
• отдАм
• тебЕ
• Этот
• подАрок

A few notes:

• отдам is stressed on the second syllable: отдАм
• тебе is stressed on the last syllable: тебЕ
• подарок is stressed on да: подАрок

If learners ignore stress, they may still be understood sometimes, but correct stress makes your Russian sound much more natural.

Yes. Подарок can often be translated as either gift or present.

Which English word sounds better depends on style, not grammar:

• gift = a bit more neutral or formal
• present = often more everyday and conversational

So the Russian sentence can match either very naturally.

Sometimes, yes. Ужин is usually translated as dinner or supper, depending on the speaker’s variety of English and eating schedule.

The Russian word itself just refers to the evening meal.
So if the meaning has already been given to the learner as after dinner, that is perfectly natural, but after supper could also fit in some contexts.