Learn Hardcore German: Vor dem Tor des Gartens brennt eine alte Laterne. - In front of the garden gate an old lantern is burning.

In German, the preposition vor (“in front of”) takes the dative case when it indicates a static location. Tor is a neuter noun (das Tor), so its dative singular form is dem Tor.

Why is des Gartens in the genitive case?

Here des Gartens expresses possession (“the gate of the garden”). German uses the genitive to show that relationship. Garten is masculine (der Garten), so its genitive singular is des Gartens.

What’s the difference between dem Tor and des Gartens?

They play two different roles:

dem Tor is the object of the preposition vor (dative case).
des Gartens is a genitive attribute showing that the gate belongs to the garden (“gate of the garden”).

Why does the verb brennt come after the prepositional phrase and before eine alte Laterne?

German main clauses follow the V2 rule: the finite verb must be the second element. Starting the sentence with Vor dem Tor des Gartens makes that phrase the first element, so brennt occupies the second position, followed by the subject.

Why does the adjective in eine alte Laterne end with -e instead of -en?

With an indefinite article in the nominative feminine singular, you use a weak adjective ending -e. The pattern is:
eine (indefinite article) + alte (adjective-weak) + Laterne (noun).

Why is Laterne feminine?

Many German nouns ending in -e are feminine. The gender of each noun must be learnt, but as a rule of thumb, Laterne (lamp) is die Laterne.

Could you replace Vor dem Tor des Gartens with Am Tor des Gartens? Would that change the meaning?

Yes.

am Tor (an dem Tor) means “at the gate” (right next to or attached to it).
vor dem Tor means “in front of the gate” (a little bit away from it).
They describe slightly different spatial relationships.

Can I use a compound noun instead of Tor des Gartens?

Absolutely. German loves compounds. Gartentor (“garden gate”) is common. You could say:
Vor dem Gartentor brennt eine alte Laterne.

Why are all nouns capitalized in this sentence?

In German, every noun (not just proper names) must be capitalized. That’s why Tor, Garten, and Laterne all start with a capital letter.

Why use the verb brennen for a lantern instead of leuchten or scheinen?

brennen (“to burn”) is used for light sources that literally burn fuel (e.g., oil lamps, candles).
leuchten and scheinen (“to shine”) focus on the emission of light and are often used for electric lights or celestial bodies.
Since Laterne implies a burning lamp, brennen is the natural choice.