Learn Hardcore German: Unser Coach zeigt uns eine Atemübung, die wir auch auf dem Gipfel machen können. - Our coach shows us a breathing exercise that we can also do on the summit.

Wir = subject form (nominative): wir gehen, wir lernen
Uns = object form (accusative or dative): Er sieht uns (accusative), Er hilft uns (dative)

The verb zeigen is used like this:
jemandem etwas zeigen = to show somebody something

So in the sentence:

Unser Coach = subject (who shows)
uns = indirect object (to whom he shows it)
eine Atemübung = direct object (what he shows)

Because uns is the person receiving something, it has to be dative, and the dative form of wir is uns.

How does zeigen work here? Why is it zeigt uns eine Atemübung and not some different order or case?

Zeigen normally takes two objects:

Dative for the person: jemandem
Accusative for the thing: etwas

Pattern:
jemand zeigt jemandem etwas
= someone shows somebody something

In the sentence:

Unser Coach (nominative, subject)
zeigt (verb)
uns (dative, indirect object = to us)
eine Atemübung (accusative, direct object = a breathing exercise)

Word order can vary a little (for example, Unser Coach zeigt eine Atemübung uns is grammatically possible but sounds unnatural).
The default and most natural order is: subject – verb – indirect object – direct object.

Why is it Unser Coach and not Unseren Coach?

The form of the possessive unser depends on the case, gender, and number of the noun it describes.

Coach is masculine and is the subject of the sentence, so it is nominative masculine singular.
The nominative masculine singular form of unser is unser (without an ending).

Comparison:

Nominative, masculine (subject): unser Coach ist hier
Accusative, masculine (object): Wir sehen unseren Coach

So in this sentence, since Coach is the subject, the correct form is Unser Coach, not Unseren Coach.

What exactly is eine Atemübung, and why is it eine and not ein?

Atemübung is a compound noun:

der Atem = breath
die Übung = exercise, practice

In German, the last part of a compound decides the gender, so:

die Übung → die Atemübung (feminine)

Because Atemübung is feminine and singular, the indefinite article in the accusative is:

eine Atemübung (feminine accusative)

If it were a masculine noun, you would use einen; for neuter, ein.

Why is there a comma before die, and what kind of clause is die wir auch auf dem Gipfel machen können?

The comma introduces a relative clause.

A relative clause gives extra information about a noun that comes right before it.
It is introduced by a relative pronoun (here: die) and has its own verb at the end.

Here:

Main clause: Unser Coach zeigt uns eine Atemübung
Relative clause: die wir auch auf dem Gipfel machen können

The relative clause describes Atemübung and tells you which exercise it is: the one that we can also do on the summit. German always separates a relative clause from the main clause with a comma.

Why is the relative pronoun die here and not der, das, or sie?

The form of the relative pronoun is determined by:

The gender and number of the noun it refers to
The case it has inside the relative clause

Step by step:

The noun being referred to is Atemübung:
- Gender: feminine
- Number: singular
Inside the relative clause die wir auch auf dem Gipfel machen können, the thing being done is the exercise.
- We do the exercise → the exercise is the direct object of machen, so accusative.

For feminine singular:

Nominative: die
Accusative: die

So the correct relative pronoun in this case is die.
Sie would be a personal pronoun, not a relative pronoun, and der/das do not match feminine Atemübung.

How does the word order work in die wir auch auf dem Gipfel machen können?

This is a subordinate clause (a relative clause), so German sends the verbs to the end.

Inside the clause we have:

wir = subject
auch = focusing particle (also)
auf dem Gipfel = prepositional phrase (location)
machen können = verb cluster at the end (main verb + modal)

Order:

Relative pronoun first: die
Subject: wir
Other elements (adverb, place): auch auf dem Gipfel
Verbs at the very end: machen können (main verb before modal)

So: die – wir – auch – auf dem Gipfel – machen können.
In subordinate clauses with a modal verb, the pattern is typically … [other stuff] + main infinitive + modal infinitive at the end.

Why is it machen können and not können machen at the end?

In a subordinate clause (like a relative clause or a weil-clause), all verbs go to the end. With a modal verb:

The main verb is in the infinitive.
The modal verb is also in the infinitive.
The usual order is: main verb + modal.

So:

Main clause: Wir können die Übung machen.
Subordinate: … dass wir die Übung machen können.

That is why the sentence has machen können at the end, not können machen.

What does auch do in die wir auch auf dem Gipfel machen können? Where can it go?

Auch means also or too and highlights that this is an additional possibility.

In die wir auch auf dem Gipfel machen können, auch modifies the whole idea of doing the exercise on the summit as an extra option (not just down below, but also there).

Other grammatically possible positions:

die wir auf dem Gipfel auch machen können
die wir auch auf dem Gipfel machen können

In practice, both of these are fine and natural; the choice mainly affects very slight emphasis. The version in the sentence is standard and sounds very natural.

Why is it auf dem Gipfel and not auf den Gipfel?

Auf is a two-way preposition; it can take either:

Dative → for location (where something is)
Accusative → for direction/motion (where something is going to)

Here we are talking about where we can do the exercise (location), not moving there:

Location: auf dem Gipfel (on the summit) → dative
Direction: auf den Gipfel (onto the summit, to the summit) → accusative

So auf dem Gipfel machen = do it while being on the summit, not while going there.

Why is the article dem with Gipfel? What gender is Gipfel?

Gipfel (summit, peak) is masculine in German:

Nominative singular: der Gipfel
Dative singular: dem Gipfel

Because the preposition auf here takes the dative (location), you get:

auf dem Gipfel

If it were accusative (direction), it would be:

auf den Gipfel

Why is the tense zeigt (present) and machen können (present) even though it might refer to the future?

German often uses the present tense to talk about the near future, especially when the context clearly points to a future time.

In this sentence, the situation is something like:

Now: the coach is showing the exercise.
Later: we can also do it on the summit.

Instead of using a complicated future form, German typically just uses the present:

Er zeigt uns jetzt eine Übung, die wir später auch machen können.

Future forms like wird zeigen, werden machen können exist but are used less often than English will. The plain present is usually enough when context makes the time clear.