Correct spelling, explanation: admitted is a word of Latin origin and it appeared in Middle English. In Latin, it had the form admittere while in English the base form is to admit. However, when you add the -ed ending to the verb, you need to double the last consonant, which results in the form admitted.

Definition of admitted:
past and past participle verb form – to reluctantly agree with somebody; to be allowed to enter some place; to be taken to hospital.
They both admitted that they had taken my car without my permission.
You will be admitted to the college if you pass your exams.
He couldn’t be admitted to the hospital because he didn’t have any health insurance.

Expressions with admitted:
Admitted is usually followed by -to, that, or -ing form:
The women admitted to the fraud and were taken by the police.
Kate admitted that she had taken the money, but promised that she would give it back.
She admitted to taking my car without my permission.

Incorrect spelling, explanation: as the infinitive form of admitted is to admit, many learners and native users of English make a mistake when they add the -ed ending to the verb and spell the word as admited. It is wrong, though, as you need to double the consonant -t as there is one vowel before it.

Incorrect spelling