Correct spelling, explanation: the word apart comes from Latin ad partem meaning to the side. Then it appeared in Old French as a part and since the 14th century, it has been used by people speaking English. As we can see, there is only one consonant p, not two. Therefore, apart is spelled correctly.

Definition of apart:
adverb meaning separately in place or time; isolatedly,
My husband and I were apart, which made us very unhappy.
They moved apart three years ago.

Phrases with apart:
take someone apart – when someone takes you apart, he takes you on one side to be with you alone,
The teacher took me apart to give me some advice.
take something apart – place something separately in pieces,
My husband took the clock apart when it didn’t work.
apart from – independently of,
She had a very happy life, apart from her work.
be poles apart – be completely different; opposite;
Sometimes theory and reality can be poles apart.

Incorrect spelling, explanation: appart is spelled incorrectly because of the origin of the word. It is derived from Old French as a part where there is only one p. Users of English can double consonant p because in some English words such changes take place. But here there is no point in writing two letters p. Thus, appart is incorrect and learners of the language should use the correct spelling of this word as apart.