Learn Hardcore Greek: Η δασκάλα κάνει μια απλή εξήγηση ώστε να καταλάβουμε όλοι τον νέο κανόνα. - The teacher gives a simple explanation so that we all understand the new rule.

Η is the feminine singular definite article in Greek, so η δασκάλα literally means the (female) teacher.
Greek normally uses the definite article much more than English, even when English might say a teacher in context.
So depending on the context, η δασκάλα can be translated as either the teacher or simply the (female) teacher in a general sense (e.g. “our teacher”).

What is the verb κάνει here, and why not just use εξηγεί (she explains)?

Κάνει is the 3rd person singular present of κάνω (to do / to make).
The phrase κάνει μια απλή εξήγηση is a light-verb construction: literally she makes a simple explanation.
You could also say η δασκάλα εξηγεί απλά τον νέο κανόνα (the teacher explains the new rule simply), but κάνει μια απλή εξήγηση sounds a bit more like “she gives a simple explanation” and puts stylistic emphasis on the noun εξήγηση (explanation).

What does μια απλή εξήγηση show about gender and agreement?

μια = feminine singular indefinite article (like a / one).
απλή = feminine singular adjective (simple).
εξήγηση = feminine singular noun (explanation).

All three agree in gender (feminine) and number (singular). The normal order with an indefinite article is article + adjective + noun: μια απλή εξήγηση.

What exactly does ώστε να mean? Is it like για να?

Ώστε να introduces a clause of purpose or intended result and is usually translated as so that, in order that, or so as to.
Για να also introduces purpose and is more common in everyday speech, often translated as to / in order to.
In this sentence, ώστε να καταλάβουμε όλοι = so that we all (can) understand.
Ώστε να can sound a little more formal or explicit about the result than για να, but in many contexts they overlap.

Why is the verb καταλάβουμε used instead of καταλαβαίνουμε?

Καταλάβουμε is the aorist subjunctive, 1st person plural of καταλαβαίνω (to understand).
Greek often uses the aorist subjunctive after particles like να, για να, ώστε να when the action is seen as a single, complete event (here: coming to understand the rule once).
Καταλαβαίνουμε is present indicative and would describe a continuous or repeated action (we understand / we are understanding), which is not the focus here.

Is καταλάβουμε definitely a subjunctive form? How is it built?

Yes. Καταλάβουμε is the 1st person plural aorist subjunctive of καταλαβαίνω.

Aorist stem: κατάλαβ- / καταλάβ-
Subjunctive ending for 1st plural: -ουμε
So: καταλάβ-ουμε.
It always appears after να, για να, ώστε να, etc., not by itself as an independent finite verb.

Why is όλοι (all) placed after καταλάβουμε and before τον νέο κανόνα?

Word order in Greek is flexible, but the default here is:
να καταλάβουμε όλοι τον νέο κανόνα = so that we all understand the new rule.
Όλοι modifies the subject εμείς (we – implied in καταλάβουμε), so it stays in the “subject area” of the clause.
You could also say να τον καταλάβουμε όλοι (slight change in emphasis), but όλοι still refers to the subject we.

Why is it όλοι and not όλους?

Όλοι is the nominative plural masculine form, used for the subject of the verb.
Here, the subject of καταλάβουμε is we all, so όλοι must be in the nominative.
Όλους would be accusative plural and would mean all (of them) as an object, which is not the meaning here.

Why is τον νέο κανόνα in the accusative case?

Τον νέο κανόνα is the direct object of καταλάβουμε (to understand).
Greek marks direct objects with the accusative case.

τον = masculine singular accusative article
νέο = masculine singular accusative adjective
κανόνα = masculine singular accusative noun
So they all agree in case, gender, and number.

Why is the adjective inside τον νέο κανόνα placed between the article and the noun?

The standard order in Greek for an adjective that directly modifies a noun is:
article + adjective + noun.
So τον νέο κανόνα = the new rule.
Other structures are possible (like ο κανόνας ο νέος, with a slightly different nuance), but the basic, neutral pattern is what you see here.

Is there any difference between νέος and καινούργιος for new in Greek?

Both νέος and καινούργιος can mean new, and in many contexts they can be swapped.
Roughly:

νέος often has a more general sense: new in time, recent, or young (e.g. νέος κανόνας = a new rule, νέος άνθρωπος = young person).
καινούργιος tends to emphasize that something is freshly acquired, unused, or brand-new.

Here, νέος κανόνας is completely natural and idiomatic.

Could the word order be ώστε όλοι να καταλάβουμε τον νέο κανόνα? Would that change the meaning?

Yes, ώστε όλοι να καταλάβουμε τον νέο κανόνα is also grammatical.
Greek allows relatively free word order, and moving όλοι earlier slightly highlights all of us.
The basic meaning stays the same: so that we all understand the new rule; the difference is only a subtle shift of emphasis.

Is ώστε να καταλάβουμε όλοι τον νέο κανόνα expressing purpose or result?

In this sentence it is mainly purpose: She makes a simple explanation so that we can all understand the new rule.
Ώστε can also introduce a result (especially without να, e.g. Ώστε καταλάβαμε όλοι… = So we all understood…), but with ώστε να plus subjunctive, the typical reading is intended result / purpose.