A 'career' is a job that you have for a long time. If you are a doctor then you have a 'career' as a doctor. The meaning of 'undertaking' is more general. It usually means something that you agree to do, or something that you must do. It comes from the verb 'to undertake'. For example: 'I undertook to answer this question. It was not an easy undertaking'. I hope this helps!