На ковре появилось пятно, поэтому я тёр его губкой.

A stain appeared on the carpet, so I rubbed it with a sponge.

Breakdown of На ковре появилось пятно, поэтому я тёр его губкой.

• я: I
• на: on
• ковёр: the carpet
• появиться: to appear

Questions & Answers about На ковре появилось пятно, поэтому я тёр его губкой.

Why is it На ковре появилось пятно and not Пятно появилось на ковре?

Both are correct. На ковре появилось пятно is a very common “existential” pattern in Russian: you start with the location (on the carpet) and then introduce the new thing that appeared (a stain). It feels like “There appeared a stain on the carpet.”
Пятно появилось на ковре is more neutral when the stain is already the topic.

Why is появилось neuter singular?

Because the subject is пятно, which is neuter singular. In the past tense, Russian verbs agree in gender and number:

• пятно (neut. sg.) → появилось
• пятна (pl.) → появились

What case is на ковре, and why is it ковре?

На + “static location” uses the Prepositional case (often called “locative” in this context).
ковёр (dictionary form, nominative) changes to ковре in the prepositional:

• на ковре = “on the carpet” (location)
  If it were movement onto the carpet, you’d use на ковёр (accusative): положил на ковёр = “put onto the carpet.”

Why does ковёр become (what’s happening to the spelling/stress)?