The original cipher disc was based on two coaxial circles.
The external one has uppercase letters and numbers from 1 to 4. It contains the plain text letters.
The inner circle has lowercase letters and can rotate around its axis. It contains the encrypted text letters.

In the original Alberti's disc, there are the 20 most used letters in latin language (which is the official Catholic church language).
The first step is to make your plain text fit into these letters, which are:

A B C D E F G I L M N O P Q R S T V X Z

Let's suppose the message is "ATTAC".

Then you have to chose a key among these letters.
In example, you can chose the "D".

Now chose a symbol in the inner circle, and rotate that disk until you simbol matches the key.
This symbol will be the initial symbol of the encrypted message.
In example, if we have this state:

A B C D E F G I L M N O P Q R S T V X Z 1 2 3 4
m s r d x z v i e g t f n k p b q o c & l y a h

we can chose "r" as initial symbol. We will then rotate the inner disc rightwards by one position, so that r goes under the key (D):

A B C D E F G I L M N O P Q R S T V X Z 1 2 3 4
h m s r d x z v i e g t f n k p b q o c & l y a

The encrypted text starts with the symbol r.

Looking at the discs, you can find the direct translation between our ATTAC message and the encryption symbols:

A -> h
T -> b
T -> b
A -> h
C -> s

So we can write rhbbhs and send it along with the key D, which is small and very easy to communicate out-of-band.

Wait... there's a problem! A letter is encrypted in the same way along the whole message, and this could lead to a quite easy decryption (i.e. based on letter frequency of natural language).
Alberti found a solution to this problem, adding the 1,2,3,4 numbers in the outer disc. In this way we can randomly choose to change the encryption scheme, let's see how.

After the encryption of the AT string, we don't want the second T to be encrypted like the first one, so we decide to make a variation.
We chose one number from 1 to 4, let's say 1.
The symbol corresponding to 1 is &. This is our new initial symbol, so we shall concatenate it to our encrypted string and rotate the inner disc until the & goes under the key we chose before (D):
Current encrypted text: rhb&

A B C D E F G I L M N O P Q R S T V X Z 1 2 3 4
h m s r d x z v i e g t f n k p b q o c & l y a
& l y a h m s r d x z v i e g t f n k p b q o c

Now we can use the new scheme to encrypt the rest of our message:

T -> f
A -> &
C -> y

The resulting encrypted message is: rhb&f&y.

TO BE CONTINUED