On which part of the landing process does the pilot begin to flare?
This answer is a bit dated since it applies to 747-100 and -200 aircraft in the 1990s. However, the same principles still apply, and I'm sure the same technique would work now on current 747s. Whether or not anybody uses this protocol now I do not know.
We flared the aircraft by the numbers without reference to peripheral vision. The use of outside vision was, of course, critical for runway alignment and any necessary crosswind corrections.
We flared with reference to the radar altimeter. If it was a 'talking' altimeter, it would announce the altitude of the main landing gear above the ground in the following increments: 100, 50, 40, 30, 20, 10, with the increments being in feet. If the the radar altimeter was an old non-talker, the flight engineer would read those increments.
Coming down the typical 3 degree glideslope on a stabilized approach, the nose would be 5 degrees up, and the fuel flow would be about 5,000 lbs per engine. When you heard the 100 foot call, that served simply as an alert. When you heard the 50 foot, you pulled most of the power and raised the nose 2 more degrees. When you heard the 10 foot call, you pulled the rest of the power while raising the nose an additional one degree, maybe a little more if you had the runway length to try for a real greaser at the risk of floating. Then you just waited. Touch down of the main gear would occur within 1 to 3 seconds. If it didn't occur within that time frame, you were floating and would want to reduce your pitch a degree or so.
These 'numbers' would be adjusted if you were making a non-stabilized approach, which we often did back then when conditions permitted because stabilized approaches used more time and fuel. Plus, a non-stabilized approach is more challenging and thus much more fun. As I understand it, they're frowned upon these days even to the point of ops specs requiring a stabilized approach.
Flare height is typically 10 ~ 30 feet, which varies with airplane size. At this point, the aircraft has already crossed the runway threshold and is above the runway. Usually, you will smoothly reduce throttle to idle as you cross the threshold (if you don't, the aircraft might climb away!), then flare. That's around 3 to 10 seconds before touchdown.
In a small GA aircraft, typically you flare when you're around 1 to 2 stories high. The speed will bleed off as you flare, the stall warning might sound ...... which is alright, because you would only stall 2 feet.
On a large airliner, flare happens a bit higher. The cockpit of a 747 or A380 is also much higher above the ground (imagine stacking two cargo containers on a trailer, then drive this on a road), which means flare happens much higher from the pilot's perspective.
Learning how to flare is one of the things which cannot be practiced in a simulator, because we need peripheral vision to judge height. I have heard seasoned airliner pilots having a hard time landing small aircraft, because they would always flare too high.
Here is a nice video that describes it well for a small airplane, worth a watch.
The flare is a key part of the landing and flaring too high can cause the plane to fall and possibly bounce.
A late flare or no flare can cause the plane to come in flat which can lead to a variety of issue that may or may not include bouncing on the nose wheel and other possibly dangerous situations.