I was watching a documentary on youtube and the pilots always say something that sounds like "B1E" when they have enough speed to take off. What exactly are they saying, and why do they say it?
What you are hearing is "V-One", written as V1.
It actually is said when they can no longer safely abort the takeoff with the remaining runway, but they still are not quite ready to takeoff.
As they get a little faster, there should then be a second callout of "Rotate" when they have achieved the required takeoff speed, and that is when they actually takeoff.
To answer the "why" part, it is said by the pilot who isn't actually flying the airplane since he has more time to monitor the speed and can say it at exactly the right time. It alerts the other pilot that it is no longer safe to abort the takeoff and that pretty much no matter what happens now, they should continue and come back to land if needed.
In my answer above, I simplified somewhat by only mentioning the available runway. There are other factors that can come into play even with a runway that is longer than needed. One example would be the speed after which the brakes can no longer stop the airplane because they would over-heat and become ineffective. This is known as the brake energy limit, and it is not safe to abort above this speed, even if you have "plenty" of runway.
V1 must be calculated or looked up prior to every takeoff, along with the other performance numbers. They vary based on the weight of the aircraft, the configuration of the aircraft, the altitude of the runway, and the weather (outside temperature, wind, etc.). Sometimes this is done via a computer and sometimes manually via lookup tables or even graphs.
For an example of the graphs, and more information about how temperature and altitude affect takeoff performance, see my previous answer here:
Does temperature affect takeoff performance?
There are also times when a pilot may increase or decrease V1 intentionally, even when it is not required. For instance, if there is wet ice on the runway it may not be safe to abort even at a very low speed, so you would be forced to continue no matter what (note that I'm NOT saying that this is a good idea). Using a lower power setting to extend engine life is another example, and you can read about it in my question here:
Why would an airplane pilot choose to intentionally use more runway than required for a takeoff?
For way more information than you probably care about takeoff performance, see my other answer here: Minimum Runway Length and V1
In closing, aircraft performance planning is a rather complicated subject and it is only one of many subjects that we must master in order to safely fly an airplane. This only just scratches the surface of it, and there is a lot more that must be considered (such as "hey, will we be able to out-climb that tree off the end of the runway?").
From the 1990s at two different airlines in 747-100 and -200 aircraft with 3-man crews. Call outs were the same regardless of whether it was the the Captain or the F.O. flying, so PF=Pilot Flying, NFP=Non Flying Pilot, and FE-Flight Engineer.
We'll assume a standard takeoff with more than adequate runway length, no runway contamination. The airplane is lined up, flaps at 10 degrees, the PF is on the brakes. The FE has rotated his seat to a position between the pilots and as far forward as possible. The PF has his hand on the top of the thrust levers. The FE has his hands positioned so that he can readily reach the bottom of the thrust levers to retard them should the PF overboost. Both pilots are primarily looking out the front windshield. The FE is watching the fuel flows, EPRs, and EGTs on the front panel.
When takeoff clearance is given, the PF starts advancing the thrust levers. The first call comes when it's obvious that all engines are powering up:
PF: "set takeoff power" The call is to the FE, but the PF keeps his hands lightly on the thrust levers in case of an abort. The FE moves the thrust levers to the desired power setting using the EPRs. He may need to jockey them a little.
FE: "power set" given when the FE is happy with the setting, although he will continue to monitor and adjust as necessary.
NFP: "80 knots" Used by only one of the airlines I flew for. They had an ops specs requirement that if you didn't have full direction control at 80 knots, you were to abort the takeoff. It was in my opinion an unneeded call, and that call plus the actions of a captain very much concerned with following the rules resulted in putting an airplane off a runway at KJFK. The airplane was totaled, only minor injuries.
NFP: "V1" The PF takes his hands off the thrust levers since an abort is no longer an option.
NFP: "rotate" The PF starts raising the nose to a pre-determined pitch attitude, more often than not 12 or 13 degrees.
NFP: "positive rate" Given by the NFP when he observes the vertical speed go positive. There were also operational noises caused by the removal of the locks that kept the gear lever from being positioned to UP when the aircraft was on the ground.
PF: "gear up" The NFP positions the gear lever to the UP position.
NFP: "V2" Given when the airplane has accelerated to that speed. The PF will maintain that speed until a clean-up height, typically 1000 -1500 feet AGL.
NFP: "transition" The PF lower the nose to around 5 degrees to start accelerating.
PF: "climb power" The FE brings the thrust levers back to climb power.
FE: "power set"
PF: "flaps 5" Given when the aircraft has accelerated to the minimum safe airspeed for that configuration. The NFP sets the flaps
PF: "flaps 1" Again, given when the aircraft has accelerated. At flaps 1 the trailing edge flaps are up, but the leading edge devices are still out.
PF: "flaps up" This gives you a clean wing.
Bear in mind that the speeds vary with runway altitude, ambient temperature, and weight of the aircraft. Back in the dark ages of the 1990s these speeds were grease-penciled in on a plastic card by the FE. The card was then handed to the F.O. for verification, and then placed at the front of the thrust level console in easy view. Then the two pilots would position 5 plastic bugs around the perimeter of the airspeed indicator to those speeds.
Also bear in mind that this is an 76 year-old man relating from memory that which he last did 16 years ago. He did do it a lot of times, though.