Jeg lægger fødselsdagsgaven på bordet.

Because lægge means to lay / to put something down, while ligge means to lie / to be lying.

In this sentence, the subject is doing an action to an object:
Jeg lægger fødselsdagsgaven på bordet = I put the birthday present on the table.

A useful contrast:

• Jeg lægger bogen på bordet = I put the book on the table.
• Bogen ligger på bordet = The book is lying on the table.

So:

• lægge = action, usually with a direct object
• ligge = state/location, no object being placed

Lægger is the present tense of lægge.

Danish present tense is often formed with -er:

• lægge → lægger
• komme → kommer
• tale → taler

So jeg lægger means I am putting or I put, depending on context. Danish present tense can cover both ideas that English often separates into simple present and present continuous.

Because fødselsdagsgaven is the definite form: the birthday present.

In Danish, the definite article is often attached to the end of the noun instead of being a separate word.

• en fødselsdagsgave = a birthday present
• fødselsdagsgaven = the birthday present

So the ending -n here works like English the.

It is a compound noun plus the definite ending.

Breakdown:

• fødsel = birth
• dag = day
• fødselsdag = birthday
• gave = gift/present
• fødselsdagsgave = birthday present
• fødselsdagsgaven = the birthday present

Notice the -s- in the middle: fødselsdag + gave → fødselsdagsgave.
This linking -s- is very common in Danish compounds.

English usually writes these as separate words, but Danish often combines them into one word.

Bordet means the table, while et bord means a table.

This is the same definite/indefinite contrast as with gaven:

• et bord = a table
• bordet = the table

So:

• på et bord = on a table
• på bordet = on the table

The noun bord is a neuter noun in Danish, which is why its indefinite form uses et.

Because Danish nouns belong to different grammatical genders, and that affects their forms.

There are two genders in modern Danish:

• common gender → usually takes en in the indefinite form
• neuter gender → usually takes et in the indefinite form

Examples from the sentence:

• en gave → gaven
• et bord → bordet

So the definite ending depends on the noun’s gender:

• common gender often gets -en
• neuter often gets -et

På usually means on.

So på bordet means on the table.

This is the natural preposition when something is placed on top of a surface.
Compare:

• på bordet = on the table
• i skuffen = in the drawer
• under stolen = under the chair

So in this sentence, på shows the destination/location where the present is being placed.

The word order is the basic Danish pattern:

Subject + verb + object + adverbial/prepositional phrase

So:

• Jeg = subject
• lægger = verb
• fødselsdagsgaven = direct object
• på bordet = prepositional phrase

This is very similar to English:

I put the birthday present on the table.

In a simple main clause, Danish normally has the finite verb in second position, and here that gives the familiar order:

Jeg lægger ...

Yes. Danish present tense often covers both meanings.

So jeg lægger fødselsdagsgaven på bordet can mean:

• I put the birthday present on the table
• I am putting the birthday present on the table

The exact meaning depends on context.

If Danish speakers want to emphasize an ongoing action, they can sometimes use extra wording, but very often the plain present tense is enough.

In standard Danish, jeg is often pronounced more like yai or yai̯, not like the spelling might suggest.

A few useful points:

• the j sounds like English y
• the written g is often very soft or almost disappears in everyday speech
• the vowel does not sound like English egg

So learners are often surprised that jeg does not sound very close to its spelling.

A few things matter here:

• æ is a front vowel, somewhat like the vowel in English cat, but not exactly the same
• the gg does not sound like a hard English g as in go
• the word is pronounced roughly like LEH-yer or LEH-er, depending on accent and speech style

The exact Danish sound is difficult to represent in English spelling, but the key point is this:

• do not pronounce it like lag-ger
• keep the æ open and front
• the middle consonant is softened

Yes. Danish, like German, very often combines nouns into one written word.

So English:

• birthday present
• school bus
• coffee cup

often becomes one word in Danish:

• fødselsdagsgave
• skolebus
• kaffekop

This is extremely common, and learning to recognize compounds is a big part of learning Danish vocabulary.

Sometimes yes, but it changes the nuance slightly.

• lægger is used when you lay something down, especially something that ends up in a horizontal or resting position
• sætter is more like set/place, often used when something is placed upright or in position

For a present, lægger sounds very natural if you are simply putting it down on the table.
Sætter might also be possible in some contexts, but lægger is the most straightforward choice here.

So if you are unsure, lægger is a good, natural option for this sentence.