Learn Swedish: Hon sa att äpplet var moget. - She said that the apple was ripe.

Because it is describing äpplet, which is an ett word (neuter), the adjective must take the neuter form. The dictionary form is mogen. For adjectives ending in an unstressed -en, you don't simply add a -t to the end. Instead, you drop the -n and replace it with a -t, giving us moget.

Is sa a slang word? I learned that the past tense of säga is sade.

Both are correct, and sa is not slang! Sade is the traditional, formal written past tense of säga (to say). However, it is almost universally pronounced sa in spoken Swedish. Because of this, writing it as sa has become completely standard and accepted in all but the most formal texts.

If I wanted to say that the apple was not ripe, where would inte go in this sentence?

It would go right before the verb: Hon sa att äpplet inte var moget. The phrase starting with att (that) is a subordinate clause. In Swedish subordinate clauses, sentence adverbs like inte must be placed before the verb, unlike in main clauses where they usually come after.

Why is "the apple" translated as äpplet instead of ?