Er kauft eine Zigarette.

Kauft is the 3rd person singular present tense form of kaufen.

The present-tense pattern is:

• ich kaufe = I buy
• du kaufst = you buy
• er/sie/es kauft = he/she/it buys

So Er kauft matches he buys.

Because Zigarette is a feminine noun.

The indefinite article a / an changes in German depending on gender:

• ein for masculine and neuter
• eine for feminine

So:

• ein Tisch = a table
• ein Auto = a car
• eine Zigarette = a cigarette

It is in the accusative case because it is the direct object of kauft.

He is buying what?
Eine Zigarette

That makes it the thing directly affected by the verb.

In this sentence, the feminine indefinite article in nominative and accusative is the same:

• nominative: eine
• accusative: eine

So you do not see a change here, but the case is still accusative.

You mainly learn the noun together with its article:

• die Zigarette

That tells you it is feminine.

There are also useful patterns. Nouns ending in -ette are very often feminine, and many nouns ending in -e are feminine too. But the safest habit is to memorize every noun with its article.

So learn it as:

• die Zigarette
• plural: die Zigaretten

Because all German nouns are capitalized.

So in German you write:

• der Mann
• das Haus
• eine Zigarette

This is a basic spelling rule and one of the first things English speakers notice.

Only because it is at the beginning of the sentence.

Normally, the pronoun is lowercase:

• er
• sie
• wir

At the start of a sentence, the first letter is capitalized:

• Er kauft eine Zigarette.

This is different from nouns, which are capitalized even in the middle of a sentence.

In a normal German main clause, the finite verb usually goes in position 2.

Here the structure is:

• Er = position 1
• kauft = position 2
• eine Zigarette = the rest

This is one of the most important German word-order rules. German is not exactly the same as English, but in a simple sentence like this the two languages look similar.

Yes, but the finite verb still stays in second position in a main clause.

For example:

• Er kauft eine Zigarette.
• Eine Zigarette kauft er.

The second version is grammatical, but it sounds more marked and gives extra emphasis to eine Zigarette.

So the original sentence is the most neutral and natural version.

In standard German, no. With a singular countable noun, you normally need an article or some other determiner.

So:

• Er kauft eine Zigarette. = correct
• Er kauft Zigarette. = not standard German

German usually behaves more strictly than English here.

Yes. Ein / eine can mean either a / an or one, depending on context and stress.

In this sentence, eine Zigarette normally just means a cigarette.

If you strongly stress eine, it can imply one cigarette:

• Er kauft eine Zigarette, nicht zwei.

So the word is the same, but the context tells you whether it is just an article or the number one.