As I would use the terminology, the C family of languages generally corresponds to languages that follow C's general syntactic rules, such as how you represent operators (particularly = is assignment and == is equality), curly braces for blocks, and to lesser degrees, making distinction between statement/expression, or having types precede variable names.

"C-like", by contrast, is a language whose semantics follow closely to C, and in particular, the C-with-extensions languages like C++ or Objective-C are usually going to fall into this category.