Barnen satt i gräset och åt glass medan deras mamma stod vid staketet.

Barnen is the definite plural form, meaning the children.

A useful thing to know is that barn is unusual because:

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

So the indefinite singular and indefinite plural look different only because of the article or context, while the definite plural adds -en.

That -et is the definite singular ending for many neuter nouns.

Here:

• ett gräs → gräset = the grass
• ett staket → staketet = the fence

So Swedish often puts definiteness on the end of the noun instead of using a separate word like the.

Swedish often uses the simple past where English would prefer was sitting, was standing, was eating, and so on.

So:

• satt = past of sitta = sat / was sitting
• stod = past of stå = stood / was standing

In this sentence, those forms naturally describe what was going on at that time. Swedish usually does not need a separate progressive form like English does.

Because the same subject continues.

So Barnen satt i gräset och åt glass means:

• The children sat in the grass and ate ice cream

Swedish, like English, often leaves out the repeated subject when two verbs share it:

• Barnen satt ... och åt ...

instead of repeating:

• Barnen satt ... och barnen åt ...

The second version would sound unnecessarily repetitive.

Åt is the past tense of äta, meaning to eat.

This is an irregular verb, so the past tense does not look very similar to the infinitive:

• äta = to eat
• äter = eats / is eating
• åt = ate
• ätit = eaten

So åt glass means ate ice cream.

Here glass is being used as a mass noun, like ice cream in English.

So:

• åt glass = ate ice cream
• åt en glass = ate an ice cream / an ice cream cone / one serving

Also, this is a very important vocabulary point: Swedish glass means ice cream, not the material glass.

Medan means while and introduces a clause describing something happening at the same time.

So the structure is:

• Barnen satt i gräset och åt glass
• medan deras mamma stod vid staketet

That gives the idea of two simultaneous actions:

• the children were sitting and eating
• while their mother was standing by the fence

Because sin/sitt/sina refers back to the subject of its own clause, and that does not work here.

In medan deras mamma stod vid staketet, the possessive is inside the subject phrase deras mamma. Swedish does not use sin there.

A simpler way to think about it:

• sin mamma works when it refers back to a subject already outside that noun phrase, as in Barnet såg sin mamma = The child saw his/her own mother
• but in their mother stood..., the subject itself is their mother, so Swedish uses deras

So deras mamma is the correct choice.

After possessives like min, din, hans, hennes, deras, Swedish normally uses the noun without a separate article and without the definite ending.

So:

• mamma = mother
• deras mamma = their mother

Not:

• deras mamman
• deras en mamma

This is different from English only in form, not in meaning.

That is mostly about natural preposition choice.

• i gräset literally means in the grass, and Swedish often uses this for being seated among or down in the grass
• vid staketet means by / next to / at the fence

So vid is a common choice for being near something like:

• vid dörren = by the door
• vid bordet = at/by the table
• vid staketet = by the fence

You may also hear på gräset in other contexts, but i gräset gives a very natural image of sitting down in the grass.

Yes. Medan introduces a subordinate clause, and subordinate clauses follow Swedish subordinate-clause word order.

In this sentence, you do not see a big difference because there is no negation or sentence adverb:

• medan deras mamma stod vid staketet

But if you added inte, the special word order becomes clearer:

• medan deras mamma inte stod vid staketet

In a main clause, Swedish often has the finite verb earlier, but in subordinate clauses words like inte usually come before the finite verb. This is a very important Swedish pattern.