Ich übe noch einen Satz mit Dativ und einen mit Akkusativ, damit der Unterschied klar wird.

Here noch means one more / another / additionally, not just still.

So Ich übe noch einen Satz ... means something like:

• I’m practicing one more sentence ...
• I’m practicing another sentence ...

A useful pattern is:

• noch ein Buch = another book / one more book
• noch einen Satz = another sentence / one more sentence

Without context, noch can sometimes mean still, but in noch einen Satz, English speakers should usually think another / one more.

Because Satz is a masculine noun and it is the direct object of üben.

The verb üben takes an accusative object:

• Ich übe einen Satz.
• Ich übe die Grammatik.

Since Satz is masculine singular, the accusative form of ein is einen:

• nominative: ein Satz
• accusative: einen Satz

So Ich übe noch einen Satz ... is correct because the speaker is practicing a sentence.

Because Satz is omitted the second time to avoid repetition.

The full version would be:

• Ich übe noch einen Satz mit Dativ und einen Satz mit Akkusativ ...

German often leaves out a repeated noun when it is obvious from context. So:

• einen mit Akkusativ = einen Satz mit Akkusativ

This is very natural and common.

Because the case of Satz depends on its role in the sentence, not on the grammar topic being mentioned.

Here, Satz is the object of übe, so it must be accusative:

• Ich übe einen Satz.

The phrase mit Dativ only describes what kind of sentence it is:

• a sentence with dative
• a sentence with accusative

So the sentence means:

• I’m practicing one sentence that uses the dative, and one that uses the accusative ...

The noun Satz itself is still accusative because of üben.

That is a very common way to talk about grammar in German.

When naming grammar categories, German often uses the bare term without an article:

• ein Verb mit Dativ
• ein Satz im Perfekt
• ein Beispiel ohne Artikel

So mit Dativ and mit Akkusativ are natural here.

You can also sometimes hear mit dem Dativ or mit dem Akkusativ, especially when speaking more explicitly about the case as a concept. But in grammar explanations, the shorter version without the article is very common.

And yes, mit normally takes the dative. If you had an article there, it would be dative:

• mit dem Dativ

But since the article is omitted, you do not see that ending.

Here damit means so that or in order that.

It introduces a purpose clause:

• Ich übe ... , damit der Unterschied klar wird.
• I’m practicing ... so that the difference becomes clear.

This damit is a conjunction. It does not mean the literal prepositional phrase with that in this sentence.

Compare:

• Ich mache das, damit du es verstehst. = I’m doing that so that you understand it.
• Ich mache das damit. = I’m doing that with that.

So in your sentence, damit clearly means so that.

Because damit introduces a subordinate clause, and in German, subordinate clauses are normally separated by a comma.

So:

• ..., damit der Unterschied klar wird.

This comma is not optional in standard German.

English punctuation is often looser here, but German is stricter about commas with subordinate clauses.

Because damit introduces a subordinate clause, and in German, the conjugated verb goes to the end of a subordinate clause.

Main clause order:

• Der Unterschied wird klar.

Subordinate clause after damit:

• ..., damit der Unterschied klar wird.

This is one of the most important word-order patterns in German.

Other examples:

• Ich lerne, weil ich morgen eine Prüfung habe.
• Sie spricht langsam, damit alle sie verstehen.

So the final position of wird is exactly what you should expect after damit.

Because der Unterschied is the subject of wird, so it must be nominative.

The structure is:

• der Unterschied = the thing that becomes clear
• wird klar = becomes clear

So literally:

• the difference becomes clear

Since Unterschied is masculine singular, the nominative form is der Unterschied.

If it were a direct object, then you would get den Unterschied instead:

• Ich verstehe den Unterschied.

But that is a different structure.

Literally, klar wird means becomes clear.

The verb here is klar werden:

• Etwas wird klar. = Something becomes clear.

So:

• damit der Unterschied klar wird
• so that the difference becomes clear

This is a very natural German way to express the idea that something is understood more clearly. It does not mean that someone is actively making the difference clear; it means the difference becomes clear to the learner.

Because they are nouns, and all nouns in German are capitalized.

The names of grammatical cases are nouns:

• der Dativ
• der Akkusativ
• der Nominativ
• der Genitiv

So even when they are used in grammar explanations, they are written with capital letters.