Definition
How many more reps a lifter could have done at the end of a set; 2 RIR means they stopped two reps short of failure.
Same idea as RPE, counted from the other end. 0 RIR is failure and equals RPE 10; 1 RIR is RPE 9; 3 RIR is RPE 7. Some coaches find clients estimate "reps left" more easily than "effort out of ten".
Most hypertrophy work sits at 1 to 3 RIR, strength work at 1 to 2, and technique or warm-up work at 3 to 4. The last few reps before failure are the ones that drive adaptation, and RIR keeps the client near them without arriving at failure on every set.
RIR is a guess and is usually guessed high. The fix is an occasional AMRAP set with the same weight: if the client thought they had 2 left and got 5, recalibrate.