Jag lägger nyckeln i lådan.

Because nyckeln is the definite form: the key.

In Swedish, definiteness is usually added as an ending on the noun:

• en nyckel = a key
• nyckeln = the key

So in this sentence, the speaker is talking about a specific key, not just any key.

For the same reason: lådan means the box / the drawer.

• en låda = a box / a drawer
• lådan = the box / the drawer

Swedish often uses the definite ending -n with common-gender nouns:

• nyckel → nyckeln
• låda → lådan

So the sentence refers to a specific box or drawer.

Lägger is the present tense of lägga, which means to lay / to put (something down).

In this sentence, it means put.

A useful thing to know is that Swedish often chooses different verbs depending on the position of the object:

• lägga = put something so that it lies flat
• ställa = put something so that it stands upright
• sätta = put something so that it sits / place someone somewhere

So with a key, lägga is very natural, because a key normally ends up lying flat in the drawer or box.

Here i covers the idea of both in and into, depending on context.

English often distinguishes:

• in = location
• into = movement

But Swedish very often uses i for both, and the verb shows whether there is movement.

So:

• Nyckeln ligger i lådan. = The key is in the drawer.
• Jag lägger nyckeln i lådan. = I put the key into the drawer.

So in your sentence, i lådan means something like into the drawer / in the drawer, with the motion understood from lägger.

This is the normal basic Swedish word order:

subject + verb + object + place

So:

• Jag = subject
• lägger = verb
• nyckeln = object
• i lådan = place/direction phrase

That makes the sentence very straightforward and natural.

Yes.

Swedish present tense often covers both:

• I put
• I am putting

So Jag lägger nyckeln i lådan can mean:

• I put the key in the drawer
• I am putting the key in the drawer

The exact English translation depends on context.

The base form is lägga.

Some useful forms are:

• att lägga = to put / to lay
• lägger = put / am putting
• lade = put / laid
• lagt = put / laid

So if you look it up in a dictionary, you would search for lägga.

Because Swedish often chooses placement verbs more specifically than English does.

Very roughly:

• lägga = lay something down
• ställa = stand something up
• sätta = set something in a sitting position / place someone somewhere

A key is usually thought of as ending up lying in the drawer, so lägger is the natural choice.

Examples:

• Jag lägger boken på bordet. = I put the book on the table.
• Jag ställer flaskan på bordet. = I put the bottle on the table.
• Jag sätter barnet i stolen. = I put the child in the chair.

English uses put for all of these, but Swedish often does not.

It can mean either, depending on context.

• låda = box
• låda = drawer

So i lådan could mean:

• in the box
• in the drawer

Usually the situation makes it clear which one is meant.

A learner-friendly guide is something like NIK-keln, but with Swedish sounds.

A few points:

• ny does not sound like English new
• the y is a Swedish front rounded vowel, which can be tricky for English speakers
• the ck gives a clear k sound
• the ending -eln is pronounced fairly lightly

If you cannot produce the Swedish y yet, an approximation is fine at first, but it is a sound worth practicing early.

A rough guide is LEG-er, but with a Swedish ä sound.

Helpful points:

• ä sounds somewhat like the vowel in English bed, though not exactly the same
• the gg here sounds like a normal hard g
• the stress is on the first syllable: LÄG-ger

So it is roughly LÄG-ger.

Because Swedish usually puts the at the end of the noun instead of using a separate word.

Compare:

• English: the key

• Swedish: nyckeln

• English: the box / the drawer
• Swedish: lådan

This is one of the biggest structural differences from English. Instead of adding a word like the, Swedish often adds a suffix:

• bok → boken = the book
• bil → bilen = the car
• låda → lådan = the box / drawer

So nyckeln and lådan already include the meaning of the.