Mathematically, continuum can refer to the real line, its cardinality , or any continuous connected dense linear order. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The Continuum Hypothesis conjectures that there is no set of cardinality bigger than that of the natural numbers , but smaller than the cardinality of the set of the real numbers .
The continuum is called so because it was the first (and most prominent) continuous set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order. Since the complex numbers add the imaginary number, i, to the real line, this is one reason they have no natural linear order.