Hypertrophy is the increase in size of a muscle. In lifting, it means building muscle: muscles get bigger because, in response to training, the body builds more muscle protein than it breaks down.
How hypertrophy happens
Training provides the stimulus; building happens during recovery. Hard sets close to failure signal to the body that the existing muscle is not enough. In the hours and days afterwards, it builds more, provided there is enough energy and building material, meaning enough calories and protein.
The most important levers
The most important factor is progressive overload: over months you manage more reps or more weight. Then there is volume. In the meta-analysis by Schoenfeld, Ogborn and Krieger (2017), more weekly sets went along with more muscle growth. Many studies show good gains at about 10 to 20 hard sets per muscle per week, with diminishing returns from more sets.
The range of usable loads is wide. In research by Schoenfeld and colleagues (2017), light and heavy loads led to similar muscle growth as long as the sets went close to failure. For maximal strength, heavy loads were better.
Example: A set of 8 and a set of 15 reps count the same for hypertrophy if both are hard enough. These numbers are an example.
How fast it goes
Muscle grows slowly. Visible differences take months, not weeks. Beginners gain much faster in their first year of training than advanced lifters, and with every year of training the possible gains get smaller. A big calorie surplus does not speed this up; it mainly adds more fat.