Beam forming is, at its base, nothing more than combining the signals from different sources so as to maximize the signal and minimize the noise. This is done by having the antennas physically separated so that there is a delay between reception at all antennas. Then you introduce electrical delays to line the received signals up again. In that way, the signal adds up and is amplified by the number of receiving antennas. The noise doesn't add the same way, so the final sum has a better signal to noise ratio than the received signal from any single antenna.
The term "beam forming" comes from the idea that this summing effect acts much like having a single directional antenna with high gain pointed at the source. The directionality of an antenna is called a "beam." Since you are making a directional antenna from non-directional antennas, you are forming a beam: beam forming.
The direction the beam points is determined by the delays between the different antennas.
In an analog system, you have a set of antennas with fixed delays - your beam points in one particular direction.
You could build an analog system where you split off the signal from each antenna into multiple delays, and sum them together to get multiple beams. I think the losses and unwanted delays of the needed splitters and amplifiers would reduce the usefulness of such a system to the point where it isn't worth the effort.
A digital system is different in that you can vary the delays in software. There are no losses, so you can make as many beams as your processor can handle in realtime.
Digital systems incorporate algorithms whereby the base station tries to improve the communication with the clients by aiming beams at them. The time delays at which the signal from the clients arrive at the base station can be used to calculate the delays needed to aim the beam. Basically, the base station determines the direction to the client and aims one of its beams in that direction.