В аптеке мне дали глазные капли и капли для носа.

Because after в meaning in / at for location, Russian normally uses the prepositional case.

• аптека = pharmacy, drugstore
• в аптеке = in the pharmacy / at the pharmacy

So:

• Я в аптеке. = I am at the pharmacy.
• Работаю в аптеке. = I work in a pharmacy.

If you were talking about motion into a pharmacy, you would usually use the accusative:

• Я иду в аптеку. = I am going to the pharmacy.

Мне is the dative case of я (I).

In Russian, the person who receives something is often put in the dative:

• мне дали = they gave to me
• literally: to me they gave

Compare:

• Я = I
• меня = me
• мне = to me

So in this sentence, мне marks the recipient of the eye drops and nose drops.

Дали is the past tense plural form of дать (to give).

Russian past tense agrees with the subject in gender/number:

• дал = he gave
• дала = she gave
• дало = it gave
• дали = they gave

Here, the sentence does not explicitly say who gave them. Russian often uses plural past tense like this when the doer is unspecified or understood from context:

• Мне сказали... = They told me...
• Мне дали... = They gave me...

In English, this often means something like:

• they gave me
• I was given
  depending on style and context.

Here, the implied meaning is something like the pharmacist / the people at the pharmacy gave me...

In Russian, капли is the normal plural form of капля (drop), and medicines like eye drops or nose drops are commonly referred to in the plural.

So:

• глазные капли = eye drops
• капли для носа = nose drops

This is very similar to English, where we also usually say drops, not a drop, when talking about the medicine as a product.

Singular капля usually means one physical drop:

• Одна капля воды = one drop of water

But the medicine is commonly:

• капли = drops

Both are natural, but they are built differently.

1. глазные капли

This uses an adjective + noun pattern:

• глазной = ocular / eye-related
• глазные капли = eye drops

The adjective глазные agrees with капли in number/case/gender.

2. капли для носа

This uses a noun + для + genitive pattern:

• для = for
• носа = genitive of нос
• капли для носа = drops for the nose

Russian often allows both kinds of expressions. In real life, both глазные капли and капли для носа are common set phrases.

You may also hear:

• носовые капли = nasal drops

But капли для носа is especially transparent and common.

Because для requires the genitive case.

• нос = nose
• носа = of the nose / for the nose

So:

• для носа = for the nose

Other examples:

• для детей = for children
• для работы = for work
• для друга = for a friend

Whenever you see для, expect the next noun to be in the genitive.

Here, глазные капли is in the accusative case because it is the direct object of дали.

However, with inanimate plural nouns, the accusative plural is usually the same as the nominative plural. So the form looks unchanged:

• nominative: глазные капли
• accusative: глазные капли

That is why it looks like the dictionary form, even though its role in the sentence is the direct object.

The same applies to капли для носа here.

Russian has no articles like English a/an/the.

So:

• в аптеке can mean in a pharmacy or at the pharmacy
• глазные капли can mean eye drops, the eye drops, or some eye drops

The exact meaning depends on context.

Russian expresses definiteness in other ways, such as context, word order, intonation, or extra words like:

• эти = these
• какие-то = some
• те = those

But in this sentence, no article is needed.

Yes, Russian word order is more flexible than English word order because case endings show grammatical roles.

This sentence begins with В аптеке to set the scene: At the pharmacy...

Then comes мне: to me...

Then the verb: дали

So the order feels natural as:

• At the pharmacy, they gave me...

Other orders are possible, for example:

• Мне дали в аптеке глазные капли и капли для носа.
• Глазные капли и капли для носа мне дали в аптеке.

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

• В аптеке... emphasizes the place first.
• Мне дали... emphasizes the fact that I was given something.
• Глазные капли... emphasizes what was given.

Yes, very often.

Russian commonly uses an unspecified they construction where English might prefer either:

• they gave me or
• I was given

So this sentence could be understood as:

• At the pharmacy, they gave me eye drops and nose drops.
• At the pharmacy, I was given eye drops and nose drops.

Both work. The Russian sentence itself is active in form, but the subject is left unstated.

The dictionary form is глазной.

In the sentence, it appears as глазные because it must agree with капли.

Agreement in Russian means adjectives change form depending on the noun:

• глазной — masculine singular
• глазная — feminine singular
• глазное — neuter singular
• глазные — plural

Since капли is plural, the adjective is also plural:

• глазные капли

Аптека normally means pharmacy, chemist's, or drugstore/pharmacy counter, depending on the variety of English you use.

It is the standard word for a place where you get medicines.

Related words:

• аптекарь = apothecary/pharmacist (older-fashioned)
• фармацевт = pharmacist / pharmaceutical worker

So in this sentence, в аптеке simply means at the pharmacy.

Then Russian would usually mention just that one item, still often in the plural because the medicine is called капли:

• В аптеке мне дали глазные капли. = At the pharmacy, they gave me eye drops.
• В аптеке мне дали капли для носа. = At the pharmacy, they gave me nose drops.

If you literally wanted to say one drop, that would be different:

• одну каплю = one drop

But for the medicine/product, Russian normally keeps капли in the plural.