Definition:
No scientific definition – based on “how many repetitions are left in the ‘TANK’?”
Relationship with RPE:
RIR is in close relationship with RPE for example:
If you do a bench-press exercise and athlete rates exercise intensity at 8/10 on the RPE scale, thus indicates 2 RIR.
RIR = 10 – RPE
You can use one or the other.