Basically anyone can carry out a UK company formation and become a company director unless they have been specifically banned from doing so (usually as a result of bankruptcy or legal proceedings). There is an age limit to form a limited company. You have to be 16 years of age or older (Section 157 Companies Act 2006). Although in practical terms it is extremely difficult to do things like open a bank account for your company until you are 18.
You only need one person to form a limited company. According to Section 154 of the Companies Act 2006 - A company limited by shares requires at least one (human) director but there may be more and it is possible for a limited company to be director of another limited company, as long as there is at least one human being who is already a director.
There is no longer a requirement to have a company secretary (Section 270 The Companies Act 2006) but you may choose to do so if you wish.
The final requirement is that you have, or can provide, a UK address to register to company to (this is something we can help with).