Paidassa on vielä tahra, vaikka pesin sen eilen.

Because -ssa/-ssä is the inessive case, which usually means in or inside.

• paita = shirt
• paidassa = in the shirt / on the shirt

With clothes, Finnish often uses this case for something located on the garment, so Paidassa on tahra is the natural way to say that the shirt has a stain.

Also, the word stem changes:

• paita → paida- → paidassa

Finnish often uses a location + olla structure where English uses have.

So instead of saying The shirt has a stain, Finnish says something closer to:

• In/on the shirt there is a stain

This is a very common pattern in Finnish. It focuses on where something is located.

Here tahra is the thing that exists in the existential sentence Paidassa on tahra.

In this kind of sentence, a singular countable thing is usually in the nominative:

• Paidassa on tahra = There is a stain on the shirt.

Compare:

• Paidassa on tahroja = There are stains on the shirt.
• Paidassa on likaa = There is dirt on the shirt.

So tahra is in the basic form because it is a single countable item being stated to exist there.

Vielä means still here.

It shows that the stain remains even after the washing:

• Paidassa on vielä tahra = There is still a stain on the shirt.

So the idea is not just that there is a stain, but that the stain is there despite what happened before.

Vaikka means although, even though, or even if, depending on context.

Here it introduces a contrast:

• vaikka pesin sen eilen = although I washed it yesterday

So the full sentence sets up a contrast between:

• washing the shirt yesterday
• the stain still being there now

After vaikka, Finnish uses a normal finite clause. You do not need a special verb form.

Because pesin is the past tense: I washed.

• pesen = I wash / I am washing
• pesin = I washed

Since the sentence says eilen (yesterday), the verb has to be in the past tense.

The verb is pestä = to wash, and pesin is its first-person singular past form.

This is about the object case.

In pesin sen, sen is used because the shirt is treated as a total object: the washing is seen as a completed whole.

• pesin sen = I washed it / I washed the whole thing
• pesin sitä = I was washing it / I washed it for a while / the action is incomplete or not viewed as a whole

So sen fits naturally here because the speaker means they washed the shirt as a complete action yesterday.

Sen refers back to paidassa — in other words, to the shirt.

Finnish often uses a pronoun like this instead of repeating the noun:

• Paidassa on vielä tahra, vaikka pesin sen eilen.

Literally:

• There is still a stain in/on the shirt, although I washed it yesterday.

So sen = it, meaning the shirt.

Finnish does not have articles like English a/an and the.

So:

• paita can mean a shirt or the shirt
• tahra can mean a stain or the stain

Context tells you which meaning is intended. In this sentence, the meaning is naturally understood from the situation.

No, Finnish word order is fairly flexible, although some orders sound more neutral than others.

This sentence is a natural neutral version:

• Paidassa on vielä tahra, vaikka pesin sen eilen.

You could also say:

• Vaikka pesin sen eilen, paidassa on vielä tahra.

That puts more focus on the contrast from the start: Even though I washed it yesterday...

So the word order can change, but the emphasis changes too.