The standard circuits $AC^i$, $NC^i$ are constructed using $AND$, $OR$ and $NOT$ of various fan-ins, fan-outs and depths.
What is the comparator gate constituted from?
Structurally why is it believed $NC^i$ is not in any $CC^j$ if $i\geq 1$ and why is it believed $CC^i$ is not in any $NC^j$ if $i\geq 1$ (refer https://arxiv.org/abs/1208.2721 for the conjecture $NC$ and $CC$ are incomparable)?
A comparator circuit is a circuit computing a function on $x_1,\ldots,x_n \in \{0,1\}^n$. The inputs to the circuit are labeled with either constants ($0$ or $1$), inputs ($x_1,\ldots,x_n$) or negated inputs ($\lnot x_1,\ldots,\lnot x_n$). The only allowed gate is a comparator gate, which has two inputs $a,b$ and two outputs $a\land b,a\lor b$. The circuit has no fan-out, that is, an output cannot be duplicated. One of the "wires" is designated as an output (just like a usual circuit).
Equivalently, we can represent a comparator circuit as a "straight-line program" with $m$ variables $z_1,\ldots,z_m$, each initialized by a constant, an input, or a negated input. The program itself is a sequence of instructions of the form $$ z_i,z_j \gets z_i \land z_j,z_i \lor z_j. $$ One of the variables $z_o$ is the output of the circuit.
It is believed that CC and NC are incomparable since nobody managed to show that one of them is contained in the other. Usually, such inclusions are either easy or false, though sometimes this intuition is wrong (the best example is SL=L).
