Barnen springer på stigen vid sjön.

Because barnen is the definite plural form of barn.

The forms are:

• ett barn = a child
• barnet = the child
• barn = children
• barnen = the children

So in this sentence, barnen means the children.

A useful detail: barn is one of those Swedish nouns whose indefinite singular and indefinite plural look the same.

Springa is the infinitive form, meaning to run.

In a normal sentence like this, Swedish uses a finite verb, so you need the present tense form:

• att springa = to run
• springer = run / are running

So Barnen springer means The children run or The children are running, depending on context.

Because Swedish verbs do not change for person or number in the present tense.

So you get:

• jag springer = I run
• du springer = you run
• han/hon springer = he/she runs
• vi springer = we run
• de springer = they run

This is much simpler than English, where you have forms like run and runs.

Stigen is the definite singular form of stig, which means path or trail.

The forms are:

• en stig = a path
• stigen = the path

Swedish usually adds the definite article to the end of the noun instead of using a separate word like the.

So:

• på stigen = on the path / along the path

If it were indefinite, it would be:

• på en stig = on a path

Here, på is best understood as on or along.

With things like paths, roads, and streets, Swedish often uses på where English may say either:

• on the path
• along the path

So springer på stigen means the children are moving on/along the path.

It does not necessarily mean they are standing on top of something in a very literal way. It is just the normal preposition for being or moving on a path, road, street, etc.

Vid usually means by, near, or beside.

So vid sjön means by the lake or near the lake.

This is different from other prepositions:

• i sjön = in the lake
• på sjön = on the lake
• vid sjön = by the lake / near the lake

So vid is the natural choice if you mean location next to the lake rather than inside it or out on the water.

Because Swedish usually marks definiteness by adding an ending to the noun.

For sjö:

• en sjö = a lake
• sjön = the lake

So Swedish often does in one word what English does with two words.

This is true for many nouns:

• en bok → boken
• en stig → stigen
• en sjö → sjön

One important extra point: when there is an adjective, Swedish often uses double definiteness:

• den stora sjön = the big lake

But in your sentence there is no adjective, so sjön by itself is enough.

The sentence follows a very normal Swedish main-clause pattern:

• Barnen = subject
• springer = verb
• på stigen vid sjön = place information

So the structure is basically:

Subject + Verb + Adverbial

A very important rule in Swedish is that the finite verb is usually in second position in a main clause. This is often called the V2 rule.

That matters more clearly if something else comes first:

• Vid sjön springer barnen på stigen.
• På stigen vid sjön springer barnen.

In both cases, the verb springer still stays in second position.

It can mean either, depending on context.

Swedish present tense often covers both:

• The children run
• The children are running

Unlike English, Swedish does not usually need a special be + -ing form for an action happening right now.

So:

• Barnen springer can mean The children are running
• It can also mean The children run, if you are talking more generally

In a sentence describing a scene, English would often translate it as are running.