Circuit

From a user perspective, a circuit is like a program, which can be written in a programming language like Go. However, internally a circuit is a constraint system; that is, a list of constraints which have an algebraic form.

For Groth16, a constraint looks like $(\sum_ia_ix_i)(\sum_ib_iy_i)=\sum_ic_iz_i$ where $a,b,c$ are constants and $x,y,z$ are variables which depend on the secret inputs known by a prover.

Translating a circuit, written with gnark API to such a constraint system is called the "arithmetization" of a circuit.

An important point is that every component of a constraint (variables, inputs and constants) live in $\mathbb{F}_p$, a finite field of characteristic $p$. To write a circuit which contains a reasonable number of constraints, it is important to work on the field $\mathbb{F}_p$, so that the field in which the circuits variables live is the same as the field on which the constraint system reasons.

On the other hand, a circuit reasoning on variables which live in $\mathbb{F}_r$ where $r\neq p$, has a high number of constraints because of the algebraic constraints needed to emulate the arithmetic modulo $r$ on a field of characteristic $p$.

Finally, the number of constraints in a circuit is limited; you cannot write arbitrarily large circuits. For example, using Groth16 on BN254, you cannot exceed ~$250M$ constraints.