Correct spelling, explanation: the word career comes from Latin as carrus, referring to a chariot. In Middle French, the word was known as carriere and then since the sixteenth century the second consonant r disappeared and the word has been used as career. The form is spelled in this way nowadays. The users of English can confuse it with the word in another language, for example in Italian this is carriera, and in German Karriere. Nevertheless, in English, there is only one r and the word should be spelled as career, not carreer.

Definition of career:
noun/verb – progress through your life; the variety of experiences and training programs that help you advance in pay or responsibility,
Anne had a very successful career in social work.
John left his recent pie to pursue an acting career.

Phrases with career:
brilliant career – very successful working life,
When my mum retired, she was able to look back on a brilliant career.
career girl/woman – her main aim in life is to achieve success in her profession,
She worked hard but she was never a career woman.

Incorrect spelling, explanation: carreer is a misspelling or a typo. The learners of English can confuse it with another word carrier, referring to a person or company that transports passengers or goods for hire. But career has a completely different meaning. Admittedly, the word originates from Latin as carrus with two r but, over time, the double r disappeared and now the correct form is career. Hence, carreer is wrong.

Incorrect spelling

Incorrect spelling

Incorrect spelling