Is there a cipher that uses part of the already-computed ciphertext in the encryption scheme? Are there any famous ciphers that, to determine the encrypted letter at position n in the ciphertext, it uses the encrypted letter at position n-1 in the ciphertext?

I tried google but I couldn't figure out how to phrase my search terms correctly to find something like this.
The concept you mentioned is used in self synchronising stream ciphers. For example, CFB mode of a block cipher uses
$$C_0=IV$$
$$C_n=E_K(C_{n-1})\oplus P_n, \quad n \geq 1$$
with $C_n,P_n$ denoting the $n^{th}$ plaintext and ciphertext blocks.
So if you substitute "block" for "symbol", this is the same as what you want. CBC mode also gives a self synchronising stream cipher.

Moustique is a self synchronising stream cipher designed by J Daemen.