A project (often shortened to "proj") is any climb that takes more than one session to complete. Projects sit at the upper limit of a climber's ability — hard enough that they cannot send it first try, but feasible with enough work. Projecting is one of the core activities of intermediate-and-above climbing.
The process usually involves working sections in isolation: "I've done the lower crux but the top remains hard." Climbers refine their beta (the sequence of moves they'll use), build specific strength for the moves, find better resting positions, and learn how the climb feels under fatigue. Some projects send within a session or two; others — at the top end — take years.
Projecting separates climbers who plateau from those who progress. The willingness to attempt routes well above your current grade, fall repeatedly, and treat each session as data rather than failure is the engine of long-term improvement. Adam Ondra's 2017 send of Silence (5.15d / 9c) was the result of two years of projecting; most everyday projects are far less dramatic but follow the same principle.