Dammsugaren står i hallen, men jag behöver först en hink med varmt vatten.

Because dammsugaren is the definite form: the vacuum cleaner.

The basic noun is en dammsugare = a vacuum cleaner.
Swedish usually puts the definite article at the end of the noun instead of using a separate word like the:

• en dammsugare = a vacuum cleaner
• dammsugaren = the vacuum cleaner

So the sentence refers to a specific vacuum cleaner that both speaker and listener can identify.

Swedish forms compound nouns very freely, and they are usually written as one word.

dammsugare is historically made from:

• damm = dust
• sugare = sucker

So literally it is something like dust-sucker, which became the normal word for vacuum cleaner.

Swedish often uses position verbs where English just uses is.

Here, står comes from stå = stand. It is used because a vacuum cleaner is thought of as being in an upright standing position.

So:

• Dammsugaren står i hallen = literally The vacuum cleaner stands in the hall
• but in natural English you would usually say The vacuum cleaner is in the hall

This is very common in Swedish:

• things upright often står
• things lying flat often ligger
• things attached/seated in place often sitter

Hallen usually means the hall, the hallway, or more specifically the entry hall / entrance area in a home.

In Swedish homes, hall often refers to the space just inside the front door, not necessarily a long corridor.

So in this sentence, i hallen most likely means in the entryway or in the hall.

Because the speaker means a specific hall: the hall in the house or apartment they are talking about.

• i hallen = in the hall
• i en hall = in a hall

After a preposition like i, Swedish can still use either definite or indefinite nouns, depending on meaning. Here, definite makes sense because it is a known place.

Behöver is the present tense of behöva, which means to need.

So:

• behöva = to need
• behöver = need / am needing
• behövde = needed

In this sentence, jag behöver means I need.

Also, behöva can be followed by either:

• a noun phrase: Jag behöver en hink = I need a bucket
• an infinitive: Jag behöver städa = I need to clean

Here it is followed by a noun phrase: en hink med varmt vatten.

Here först means first or before anything else.

So the idea is:

I need a bucket of warm water first, before I do the next thing.

It expresses order or priority.

Because men is a coordinating conjunction, and after it Swedish normally keeps regular main-clause word order:

• men jag behöver först ...

That means:

• jag = subject
• behöver = finite verb
• först = adverb

If you move först to the front for emphasis, Swedish uses inversion:

• Men först behöver jag en hink med varmt vatten.

Both are correct, but they have slightly different emphasis.

Because hink is a common-gender noun, an en-word.

So:

• en hink = a bucket
• hinken = the bucket

It is also indefinite here because the speaker does not mean one particular bucket already identified in the conversation. Any suitable bucket will do.

Swedish often uses med = with where English might prefer of.

So:

• en hink med varmt vatten = literally a bucket with warm water
• natural English: a bucket of warm water

In Swedish, med describes what the bucket contains.

Because vatten is a neuter noun, an ett-word.

When an adjective describes an indefinite singular neuter noun, it usually gets -t:

• ett varmt vatten would be grammatical in form, though normally you would just say varmt vatten
• compare en varm hink = a warm bucket

So:

• varm is used with common-gender singular nouns
• varmt is used with neuter singular nouns

That is why you get varmt vatten.

Because vatten is being used as an uncountable/mass noun here.

Swedish often leaves out the article with substances and materials when speaking generally:

• vatten = water
• mjölk = milk
• kaffe = coffee

So varmt vatten simply means warm water, not a warm water.

If it were definite, you could say:

• det varma vattnet = the warm water

Not strictly. In modern Swedish, commas are often used less than in English, and the comma before men can be optional depending on style.

So both of these are possible:

• Dammsugaren står i hallen, men jag behöver först en hink med varmt vatten.
• Dammsugaren står i hallen men jag behöver först en hink med varmt vatten.

The version with the comma gives a clearer pause between the two clauses.