Since free C*-algebras don't exist, we can't give a concrete description of what relations are allowed.  Instead, we need to give axioms that determine what collections of n-tuples $(a_1,\dots,a_n)$ in $A$ are allowed, where $A$ varies over all C*-algebras.  It is important that the elements not "know about" the ambient C*-algebra, so ``a is in a separable C*-algebra'' is not allowed.

Some of what is allowed is conditions like $0 \leq x \leq 1$ and 
$$
0\leq\left[\begin{array}{cc}
\mathbf{1} & x\\
x^{*} & \mathbf{1}
\end{array}\right]\leq 1
$$
where $\mathbf{1}$ is in the unitization of $A$.  Another fun example is ``x is hermitian and has spectrum contained in the Cantor set''.  We can't use Borel functional calculus, but we can take a continuous function on $\mathbb R$ that is bounded and use as a relation "$x$ is normal and $f(x)=0$."  See section 5 of "From Matrix to Operator Inequalities" by me, Canad. Math. Bull. 55(2012), 339--350 for a use of analytic functional calculus.

I could drone on forever here, but I already did so here: ``C*-Algebra Relations'' in Mathematica Scandinavica 107, 43--72, 2010.