Я купил мазь от ушибов и маленький пластырь для царапины на ладони.

Купил is the past tense of купить, which is the perfective verb to buy. That tells you the purchase is seen as a completed action: I bought.

It also tells you the speaker is masculine grammatically. In the past tense, Russian shows gender:

• я купил = I bought (male speaker)
• я купила = I bought (female speaker)

If you used покупал, that would be the imperfective verb and would sound more like was buying / used to buy / bought (with focus on the process).

The soft sign ь does not automatically tell you the noun’s gender.

In this sentence:

• мазь is feminine
• пластырь is masculine

The soft sign mainly shows that the preceding consonant is softened in pronunciation.

This is something learners often have to memorize with the noun itself. Russian has many nouns ending in ь, and some are masculine while others are feminine.

For example:

• мазь — feminine
• дверь — feminine
• словарь — masculine
• пластырь — masculine

Because от in this meaning takes the genitive case.

With medicines and remedies, от + genitive often means against, for, or to treat something:

• лекарство от кашля = medicine for a cough
• таблетки от боли = pills for pain
• мазь от ушибов = ointment for bruises / against bruising

So ушибов is the genitive plural form of ушиб.

Not necessarily specific bruises. In Russian, plural is often used in this kind of phrase to mean a condition or type of problem in general.

So мазь от ушибов is a natural way to say ointment for bruises / for contusions as a general use.

It does not have to mean the speaker definitely has several bruises right now. It can just describe what the ointment is meant for.

Here царапины is genitive singular of царапина.

That may look confusing, because царапины can also be nominative plural. But after для, the noun must be in the genitive, so here it has to be genitive singular:

• царапина = a scratch
• для царапины = for a scratch

If you wanted for scratches in general, you would more likely say:

• для царапин

So the form is ambiguous by itself, but the preposition для tells you how to read it.

Yes, both can be translated as for, but they are used differently.

• от + genitive = against / to treat
• для + genitive = for / intended for / meant for

So:

• мазь от ушибов = ointment against bruises
• пластырь для царапины = a bandage/band-aid for a scratch

In other words, от is especially common with illnesses, pain, symptoms, and injuries being treated.
Для is broader and often means the object is intended for some purpose.

Because it is a masculine inanimate noun.

After купил, both мазь and маленький пластырь are direct objects, so they are in the accusative case. But for masculine inanimate nouns, the accusative looks the same as the nominative:

• nominative: маленький пластырь
• accusative: маленький пластырь

So the form does not change.

Compare that with a masculine animate noun, where the accusative would look different:

• я вижу маленького кота = I see a small cat

Because this is location, not movement.

With на, Russian uses:

• на + accusative for movement onto
• на + prepositional for location on

Here the scratch is located on the palm, so Russian uses the prepositional:

• на ладони = on the palm

If there were movement, you would use the accusative:

• положить на ладонь = to place onto the palm

In this sentence, пластырь most naturally means an adhesive bandage or Band-Aid-type plaster.

Russian пластырь can refer to different kinds of medical sticking patches, including:

• a small adhesive bandage
• a medicated patch
• sometimes a general sticking plaster

Because the sentence says маленький пластырь and mentions a scratch, the most natural meaning is a small bandage / Band-Aid.

Because Russian has no articles.

So Russian does not normally distinguish a small bandage and the small bandage with separate article words. The meaning comes from context.

That means:

• мазь can mean an ointment or the ointment
• маленький пластырь can mean a small bandage or the small bandage

If Russian wants to be more specific, it can use words like:

• этот = this
• тот = that
• какой-то = some / a certain

But very often, no article-like word is needed at all.