What is a rank-1 constraint system? Why not rank-2 constraint system or rank-3 constraint system?

How do rank-1 constraint systems link to circuits?
The "rank-1" specifically refers to the rank of the matrix which is produced.

As for the link to circuits, in zk-SNARKS at least, an arithmetic circuit is converted into a R1CS. Each constraint corresponds to one logic gate.

Here's a more detailed walkthrough by Vitalik Buterin: https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649