A hard set is a working set that ends close to failure, with the usual cutoff at no more than 3 reps in reserve (RIR). Hard sets per muscle per week are the most common way to count training volume for muscle growth.
What counts and what does not
Warm-up sets do not count. Neither do sets that end far from failure, for example stopping after 10 reps when 8 more would have gone. You do not need to go all the way to failure, though. A meta-analysis by Refalo et al. 2023 found no evidence that training to momentary muscular failure builds more muscle than stopping just short of it.
The rep count is secondary. According to a review by Baz-Valle et al., set count works as a volume measure as long as reps fall roughly between 6 and 20 or more. A set of 8 and a set of 15 count the same if both are hard enough.
How to count
Compound lifts hit several muscles. A common rule of thumb counts the main muscle fully and assisting muscles as half. In the analysis by Pelland et al., this counting method fit the study data best.
Example: A week with 4 sets of bench press, 3 sets of incline press and 6 sets of cable pushdowns gives 7 sets for chest and 6 + 3.5 = 9.5 sets for triceps. This is a worked example.
As a guide, about 10 to 20 hard sets per muscle per week is a common range. More important than the exact counting method is that you count the same way every week. Only then can you see whether a change in volume makes a difference.