Core and Extended levels are only referred to IGCSE Cambridge syllabuses. For A Level syllabuses, please refer: A Level, AS, & A2 Syllabus
The article below is a quick summary of the enquiry answer from Cambridge International's help page. You may read the full article here: What is the difference between IGCSE Core and Extended?
Some IGCSE subjects have a choice between Core and Extended papers. This is to take account of differing abilities and allows teachers to decide the appropriate level for their students.
The Core curriculum is within the ability range of a large majority of students. It covers the full overview of the subject and is targeted for students expected to achieve grades C to G.
The Extended curriculum is made up of both the Core curriculum and Supplement. It has been designed for the more academically able and leads naturally into higher education or professional training. It is targeted for students expected to achieve grades A to E.
If a student falls in the 'middle' of the range, the Extended level is recommended as the student may be capable of achieving the higher grades. If the student had entered for the Core curriculum, they are not able to achieve above a C.
In Step 2: Subject Selection of the exam registration, Core papers are labelled with Core, while Extended papers are labelled with Extended.
If you're unsure of the level you should choose, please consult your teacher(s).