Learn Hardcore German: Nach dem Abitur möchte sie ein Jahr im Ausland arbeiten. - After graduating from high school she wants to work abroad for a year.

You can sometimes drop articles with certain fixed phrases or with proper names (for example, nach Hause, nach Deutschland), but Abitur does not behave like that here. You need dem.

Why does the sentence start with Nach dem Abitur and then put möchte before sie?

German main clauses follow the verb-second (V2) rule: the conjugated verb must be in second position in the clause.

You can put many different elements in first position (the Vorfeld), for example:

Sie möchte nach dem Abitur ein Jahr im Ausland arbeiten.
Nach dem Abitur möchte sie ein Jahr im Ausland arbeiten.
Ein Jahr im Ausland möchte sie nach dem Abitur arbeiten.

In your sentence, the phrase Nach dem Abitur is placed first for emphasis (time comes first). Because something is already in first position, the finite verb möchte must be second, and the subject sie comes after it:

1st position: Nach dem Abitur
2nd (verb) position: möchte
then: sie … arbeiten

Could I also say Sie möchte nach dem Abitur ein Jahr im Ausland arbeiten? Is there any difference?

Yes, that is absolutely correct as well:

Sie möchte nach dem Abitur ein Jahr im Ausland arbeiten.

The main difference is emphasis / information structure:

Sie möchte … – neutral start; focus is slightly more on she and what she wants.
Nach dem Abitur möchte sie … – puts more emphasis on the time frame (what happens after the Abitur).

Grammatically, both sentences are fine and mean the same thing.

Why is arbeiten at the end of the sentence?

Möchte is a modal verb (like kann, muss, will, etc.).

In German, when a modal verb is used with another verb, the construction is:

modal verb in second position (conjugated)
the main verb in the infinitive at the end of the clause

So:

Sie möchte arbeiten.
Sie möchte im Ausland arbeiten.
Nach dem Abitur möchte sie ein Jahr im Ausland arbeiten.

This verb-final position for the infinitive is a core feature of German word order.

What form of the verb is möchte, and why is it used here?

Möchte is the Konjunktiv II (subjunctive II) form of the verb mögen.

Forms of mögen:

Present indicative: ich mag, du magst, er mag, …
Konjunktiv II: ich möchte, du möchtest, er möchte, …

In modern German, möchte is used almost exclusively to express a polite or softer wish / desire, similar to would like to in English:

Ich möchte ein Jahr im Ausland arbeiten. – I would like to work abroad for a year.

In your sentence it expresses her intention or wish in a relatively soft, polite way.

What is the difference between möchte and will in a sentence like this?

Both relate to wanting, but they differ in tone and strength:

möchte = would like to
- softer, more polite, expressing a wish or preference
- Nach dem Abitur möchte sie ein Jahr im Ausland arbeiten.
  → She would like to do that, that is her plan or wish.
will = wants to / intends to
- stronger, more direct, sometimes sounds determined or even stubborn
- Nach dem Abitur will sie ein Jahr im Ausland arbeiten.
  → She wants to; she has a firm intention.

Both are grammatically correct, but möchte sounds more polite and is more common in such planning or wish-related statements.

Why is it ein Jahr and not something like ein Jahre?

In German, after the number eins (one), the noun stays in its singular form with no plural ending:

ein Jahr – one year
ein Tag – one day
ein Monat – one month

You only use the plural form after numbers greater than one:

zwei Jahre, drei Jahre, viele Jahre

So ein Jahr is the normal, correct form for one year.

Can I say ein Jahr lang im Ausland arbeiten instead of ein Jahr im Ausland arbeiten? Is there a difference?

Yes, you can say both:

ein Jahr im Ausland arbeiten
ein Jahr lang im Ausland arbeiten

Ein Jahr lang adds a slight emphasis on the duration (for the length of one year), but in most everyday contexts the meaning is practically the same.

You can think of lang here as similar to English for:
ein Jahr lang ≈ for one year.

What exactly is das Abitur in the German system?

Das Abitur is:

the final school-leaving examination at the end of secondary school (Gymnasium)
and by extension, the qualification you get from it

It roughly corresponds to:

British A‑levels (as a university entrance qualification)
or an American high-school diploma plus standardized tests (though the systems are not identical)

So nach dem Abitur means after finishing school with the Abitur qualification.

What does im mean in im Ausland, and what case is used?

Im is a contraction of in dem:

im Ausland = in dem Ausland

Here, in is used with the dative case because it expresses location (where?), not direction (where to?).

Wo? (Where?) → im Ausland – in/abroad (dative)
Wohin? (Where to?) → ins Ausland = in das Ausland – to abroad (accusative, direction)

So im Ausland already contains the dative article dem inside im.

Why is Ausland written without an extra article like ein Ausland or something similar?

Ausland is a neuter noun (das Ausland) that often behaves like a kind of mass/abstract noun meaning foreign countries / abroad in general.

In many fixed expressions, German uses it without an additional article, especially with prepositions:

im Ausland – abroad (literally: in the abroad)
aus dem Ausland – from abroad
ins Ausland – to a foreign country / abroad

In im Ausland arbeiten, the article dem is already inside im (in dem), so you cannot or do not need to add anything else like ein.

What is the difference between im Ausland arbeiten and ins Ausland fahren?

The difference is location vs. direction and the resulting case:

im Ausland arbeiten
- im = in dem (dative)
- answers Wo? (Where?) – location
- She is located abroad and is working there.
ins Ausland fahren
- ins = in das (accusative)
- answers Wohin? (Where to?) – direction / movement
- She is going to a foreign country.

So:

im Ausland → being abroad (dative, location)
ins Ausland → going abroad (accusative, direction)

Why is Ausland capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

Since Ausland is a noun (meaning something like foreign countries / abroad), it must start with a capital letter:

das Ausland
im Ausland
aus dem Ausland

This capitalization rule applies to Abitur, Jahr, and Ausland in your sentence as well.