A digital signal is just an interpretation of an analog signal. Say you have a system of transistors that operates between 0 and 5 volts. Say the switching voltage for your logic gates is 2.5 volts. So anything over 2.5 volts will cause your transistors to switch, and anything under 2.5 will not. If this were the case, you would define anything over 2.5 volts to be a logical 1 and anything under 2.5 to be a logical zero.
This would be an ideal case. In reality, transistors do not simply switch from off to on instantaniously. There will always be some region say from 2-3 volts where it will be unclear whether the transistor is on or off. So in real systems, anything under 2 volts might be considered a zero and anything over 3 volts might be considered a 1 and anything from 2-3 volts would be undefined.
One of the reasons that digital circuits have become so much more popular than analog circuits is because they don't carry noise throughout the entire system. Say I'm trying to pass an analog signal through a multistage amplifier.
5----5x2+U ----------- (5x2+U)x2+U ------- ((5x2+U)x2+U)x2+U
-------[-------------------[-----------------------[--------------------
So the original signal is 5 volts, and each transistor amplifier has a gain of two and noise of U. As the signal propagates through the analog circuit, the noise from each transistor is amplified.
Now say we want to pass a signal through a digital circuit. The difference here is that instead of the output being tied to the input, the output is either tied to ground, or 5v, depending on whether the digital gate is off or on. For this simple example, assume that each gate is simply a transmission gate.
5v ------------- 5v+U ---- 5v----------5v+U ------- 5v------------- 5v+U -------- 5v
-------------------------[-------------------------[--------------------------------[-------------------
Here, the signal starts out at 5v and gains U noise before it reaches the first gate. If we assume that U is not enough to change the logic state of the input (ie it is less than 2v), then the first gate will consider it to be a logical 1 and connect its output to 5v. The same thing happens at every gate stage.
As long as you make sure that the noise in between each gate is not enough to change the logic state of the system, then noise will not propagate through an enitre system. So you can have a signal travel through millions of gates without it being affected significantly by noise. That would be almost impossible with even 100 analog gates...