A neef is a male relative: either a first cousin or a nephew. A nicht is a female relative: either a first cousin or a niece.
Like the word ‘cousin’ in English, the words ‘neef’ and ‘nicht’ are sometimes used to describe a more distant relationship. Someone who is referred to as a ‘neef,’ may turn out to be the husband of a first cousin once removed.
Because of the Dutch tradition to name people after relatives you will often find several ‘neven’ and ‘nichten’ with the same first (and sometimes last) name, all named after the same shared grandparent.