The core of something such as a problem or an issue is the part of it that has to be understood or accepted before the whole thing can be understood or dealt with.
A core team or a core group is a group of people who do the main part of a job or piece of work. Other people may also help, but only for limited periods of time.
You can use to the core to describe someone who is a very strong supporter of someone or something and will never change their views. For example, you can say that someone is Republican to the core .