The most crucial element is the physical, conducted by league-appointed doctors, in which players have everything (especially prior injuries) poked and prodded with X-rays and MRIs. Every year, a number of players fall unexpectedly in the draft based on the results of these medical examinations, even though it is far from an exact science and there is often quite a bit of disagreement among team doctors about the prognosis.
There are also measurements conducted to get an updated (or sometimes first) look at players' true heights, weights, wingspans and standing reaches, as well as their body-fat percentages, which can provide some insight into a player's body type and how seriously he has been training for the draft process since the season ended. Separately, players undergo a barrage of athletic testing intended to measure their speed, lateral quickness, agility, leaping ability, reaction time and strength. Teams don't put a great deal of stock into these types of tests, but some interesting information can be gleaned from the outliers on both ends of the spectrum.
Another important part of the week: 13 hours of interview sessions scheduled in three batches over three days. All 30 teams submit in advance lists of players they would like to sit down with -- a process that is reminiscent of speed dating, with players (who are typically well-prepared by their agents) going from hotel room to hotel room to meet NBA executives, coaches and team psychologists.
Finally, 40 of the players will play 5-on-5, while others will elect to participate only in shooting drills. This is a little more useful for coaching staffs and other NBA team representatives who weren't able to evaluate players live throughout the year than it is for scouts who have followed these prospects for much of their careers. But it certainly does have an impact, even if just to confirm what people already thought about prospects going into the week.
Schedule and logistics
Monday and Tuesday: Players arrive. Elite prospects conduct physicals/medical testing. Combine participants are invited to watch NBA draft lottery show.
Wednesday: Measurements. Medical testing. First batch of NBA team interviews from 2-8 p.m.
Thursday: Second batch of NBA team interviews (8:30 a.m. to noon). Shooting drills, athletic testing and competitive 5-on-5 scrimmaging (1:45-6:30 p.m.). Elite prospects depart.
Friday: Final NBA team interviews (8:30 a.m. to noon). More drills, testing and scrimmages. In the evening: Remaining combine participants meet with players association (NBPA) representatives.
Saturday: Medical testing. Meeting with NBA player development group and NBPA.
Sunday: Medical testing.
Monday: Players depart, many to NBA team facilities for private workouts.
Players who can help or hurt their stock the most
Players rarely get drafted significantly higher than they were projected off a strong week at the combine, just as a poor performance is unlikely to move the needle too far in the other direction. Significant medical red flags or an usually bad series of interviews are much more likely to affect a players' standing than a particularly hot shooting day or impressive vertical leap numbers.
That said, certain players have more to gain in Chicago than others, usually based on past exposure. Teams have extensive scouting evaluations on Devonte' Graham, who spent four years at Kansas, took his team to the Final Four and went up against most of the top prospects in the draft.