Learn Hardcore Polish: Dzieci biegają wieczorem na boisku. - Children run in the evening on the field.

There is no regular -y or -i ending added here; the whole word changes.
Also, dzieci is used for both nominative plural (subject) and accusative plural (object), so its form doesn’t change between those two cases. In this sentence, dzieci is nominative plural, the subject of the verb biegają.

Why is the verb biegają and not just biegną or something else?

Both biegają and biegną exist, but they’re different:

biegają – 3rd person plural, imperfective verb biegać (to run, habitually/regularly, in general).
biegną – 3rd person plural, imperfective verb biec (to be running right now, in a specific instance or direction).

In this sentence, Dzieci biegają wieczorem na boisku suggests a habit or repeated action:

The children run in the evening on the field (as something they often do).

If you said Dzieci biegną na boisko, it would mean The children are running to the field (now, in this moment, towards the field). Different aspect and different meaning.

What person and number is biegają, and how is it formed?

Biegają is:

3rd person plural
Present tense
Imperfective aspect
From the verb biegać

Conjugation of biegać in present tense:

(ja) biegam – I run
(ty) biegasz – you run (singular, informal)
(on/ona/ono) biega – he/she/it runs
(my) biegamy – we run
(wy) biegacie – you run (plural)
(oni/one) biegają – they run

So dzieci biegają = children run / are running (habitually).

Why is it wieczorem, not wieczór or w wieczór?

Wieczorem is the instrumental case of wieczór (evening), and with time expressions the instrumental is commonly used to mean “in the [part of the day]”.

rano (in the morning) – historically an adverb
po południu (in the afternoon)
wieczorem (in the evening)
nocą (at night)

So wieczorem literally is “in the evening” or “during the evening”.
You don’t normally say w wieczór in this general-time sense; wieczorem is the natural, idiomatic form.

What case is na boisku, and why is it not na boisko?

Na boisku uses the locative case (boisko → na boisku) because it describes a location where something happens:

na
- locative → on / at (static location)
  - Dzieci biegają na boisku. – The children run on the field.

If you used na boisko (accusative), it would mean movement toward the field:

na
- accusative → onto / to (direction, movement)
  - Dzieci biegną na boisko. – The children are running to the field.

So:

biegają na boisku = they’re running on the field (there).
biegną na boisko = they’re running to the field (towards it).

What is the dictionary form of boisku, and what does it literally mean?

The dictionary (nominative singular) form is boisko.

boisko – sports field, pitch, court (e.g. football pitch, school playground used for sports).

Declension (neuter noun):

Nominative: boisko
Genitive: boiska
Dative: boisku
Accusative: boisko
Instrumental: boiskiem
Locative: boisku ← used here after na
Vocative: (same as nominative, boisko)

In na boisku, it’s locative: on the field.

Can I change the word order? For example, can I say Wieczorem dzieci biegają na boisku? Does it change the meaning?

Yes, word order in Polish is relatively flexible. All of these are grammatically correct:

Dzieci biegają wieczorem na boisku.
Wieczorem dzieci biegają na boisku.
Na boisku dzieci biegają wieczorem.

The basic meaning stays the same. What changes is emphasis and what sounds more natural in a particular context.

Starting with Wieczorem puts more focus on when:
- Wieczorem dzieci biegają na boisku. – In the evenings, that’s when the kids run on the field.
Starting with Dzieci keeps the subject as the starting point:
- Dzieci biegają wieczorem na boisku. – It’s the children that run, and they do it in the evening on the field.

For a simple neutral statement, the original order is very natural.

Is wieczorem an adverb or a noun in a case?

Formally, wieczorem is the instrumental form of the noun wieczór.

In practice, though, it behaves very much like an adverb of time (“in the evening”), because:

It doesn’t need a preposition here.
It directly answers the question “kiedy?” (when?).

So grammatically it’s a noun in instrumental case, functionally it acts as an adverbial time expression.

What’s the difference between na boisku and na placu zabaw?

Both use na + locative for location, but they refer to different kinds of places:

na boisku – on a sports field / pitch (typically for games like football, basketball, etc.)
na placu zabaw – on the playground (with swings, slides, etc.)

So:

Dzieci biegają wieczorem na boisku. – They are running on a sports field.
Dzieci bawią się na placu zabaw. – They are playing on the playground.

Both structures are grammatically similar; the difference is lexical (what kind of place it is).

Could I say Dzieci biegają po boisku instead of na boisku?

Yes, but there is a nuance:

na boisku – simply on the field, stating the location.
po boisku – around / over the field, with a slight emphasis on movement within that area, “all over the field”.

Both are natural and correct:

Dzieci biegają na boisku. – They run on the field.
Dzieci biegają po boisku. – They run around the field (moving all over it).

In everyday speech, the difference is often subtle, and both are common.

Why is there no article like “the” or “a” before dzieci or boisku?

Polish has no articles (a, an, the). The definiteness or indefiniteness is understood from:

Context
Word order
Sometimes from additional words (like te dzieci – these children).

So Dzieci biegają wieczorem na boisku can mean:

The children run on the field in the evening.
Children run on a/the field in the evening.

English must choose a or the; Polish doesn’t mark that grammatically. The context would usually make it clear which one is meant.

How would this sentence change in the past tense and future tense?

Using the same habitual verb biegać:

Past tense (imperfective, “used to run / were running”):

Dzieci biegały wieczorem na boisku.

Future tense (imperfective, “will be running / will run habitually”):

Dzieci będą biegały wieczorem na boisku.

Note:

dzieci grammatically behave as neuter plural (even though they are people), so the verb ends in -ły in past/future compound forms: biegały, będą biegały.

If I want to say “The children are running on the field this evening (specific one time)”, would I still use biegają?

If you want to emphasize a specific, one-time event happening now or planned for a particular evening, you would more likely use:

Dzisiaj wieczorem dzieci będą biegać na boisku. – This evening the children will be running on the field.
Teraz dzieci biegną po boisku. – Right now the children are running across the field.

Biegają is more neutral/habitual:

Dzieci biegają wieczorem na boisku. – They (generally) run in the evenings on the field (habit/routine).

So for a specific planned evening, use będą biegać;
for a current ongoing run in a particular direction, use biegną.