Ciphers from the Classical Era
The classical algorithms are those invented precomputer up until around the 1950's. The list below is roughly ordered by complexity, least complex at the top.
Classical ciphers are cryptographic algorithms that have been used in the past (pre WWII). Some of them have only ever been used by amateurs (e.g. Bifid), while some of them have been used by armies to secure their top level communications (e.g. ADFGVX).
None of these algorithms are very secure as far as protecting information goes (with todays computers to break them), so if real data security is needed you should probably look at modern algorithms.

Atbash Cipher
The Atbash cipher is a substitution cipher with a specific key where the letters of the alphabet are reversed. I.e. all As are replaced with Zs, all Bs are replaced with Ys, and so on.

ROT13 Cipher
The ROT13 cipher is not really a cipher, more just a way to obscure information temporarily. It is often used to hide e.g. movie spoilers.

Caesar Cipher
The caesar cipher (a.k.a the shift cipher, Caesar's Code or Caesar Shift) is one of the earliest known and simplest ciphers.

Affine Cipher
A type of simple substitution cipher, very easy to crack.

Railfence Cipher
A simple transposition cipher.

Baconian Cipher
The Baconian cipher is a 'biliteral' cipher, i.e. it employs only 2 characters. It is a substitution cipher.

Polybius Square Cipher
The Polybius Square is essentially identical to the simple substitution cipher, except that each plaintext character is enciphered as 2 ciphertext characters.

Simple Substitution Cipher
A simple cipher used by governments for hundreds of years. Code is provided for encryption, decryption and cryptanalysis.

Codes and Nomenclators Cipher
Nomenclators are a mix between substitution ciphers and Codes, used extensively during the middle ages. Codes in various forms were used up until fairly recently.

Columnar Transposition Cipher
Another simple transposition cipher in which letters are arranged in rows and the columns are transposed according to a key.

Autokey Cipher
The Autokey cipher is closely related to the Vigenere cipher, it differs in how the key material is generated. The Autokey cipher uses a key word in addition to the plaintext as its key material, this makes it more secure than Vigenere.

Beaufort Cipher
Very similar to the Vigenere cipher, but slightly different algorithm.

Porta Cipher
The Porta cipher is a polyalphabetic substitution cipher that uses a keyword to choose which alphabet to encipher letters.

Running Key Cipher
The Running Key cipher is similar to the Vigenere cipher, but the key is usually a long piece of nonrepeating text. This makes it harder to break in general than the Vigenere or Autokey ciphers.

VigenĂ¨re and Gronsfeld Cipher
A more complex polyalphabetic substitution cipher. Code is provided for encryption, decryption and cryptanalysis.

Homophonic Substitution Cipher
The Homophonic Substitution cipher is a substitution cipher in which single plaintext letters can be replaced by any of several different ciphertext letters. They are generally much more difficult to break than standard substitution ciphers.

FourSquare Cipher
An algorithm invented by Felix Delastelle, published in 1902

Hill Cipher
An algorithm based on matrix theory. Very good at diffusion.

Playfair Cipher
The technique encrypts pairs of letters (digraphs), instead of single letters as in the simple substitution cipher. The Playfair cipher is thus significantly harder to break since the frequency analysis used for simple substitution ciphers does not work with it.

ADFGVX Cipher
A fractionating transposition cipher. Used by the Germans during the first world war, but cracked by the French. Quite a difficult cipher to break.

ADFGX Cipher
A fractionating transposition cipher. Used by the Germans during the first world war, closely related to ADFGVX (Note the extra V in the name).

Bifid Cipher
A fractionating transposition cipher. Only ever used by amateur cryptographers. Can be broken fairly easily.

Straddle Checkerboard Cipher
A substitution cipher with variable length substitutions.

Trifid Cipher
A fractionating transposition cipher. A variant of Bifid.

Base64 Cipher
Base64 isn't really a cipher, but I see it used all the time for "enciphering" text, so it gets an honorary mention.

Fractionated Morse Cipher
Fractionated Morse first converts the plaintext to morse code, then enciphers fixed size blocks of morse code back to letters. This procedure means plaintext letters are mixed into the ciphertext letters i.e. one plaintext letter does not map to one ciphertext letter.
List of Classical ciphers:
 Atbash Cipher
 ROT13 Cipher
 Caesar Cipher
 Affine Cipher
 Railfence Cipher
 Baconian Cipher
 Polybius Square Cipher
 Simple Substitution Cipher
 Codes and Nomenclators Cipher
 Columnar Transposition Cipher
 Autokey Cipher
 Beaufort Cipher
 Porta Cipher
 Running Key Cipher
 VigenĂ¨re and Gronsfeld Cipher
 Homophonic Substitution Cipher
 FourSquare Cipher
 Hill Cipher
 Playfair Cipher
 ADFGVX Cipher
 ADFGX Cipher
 Bifid Cipher
 Straddle Checkerboard Cipher
 Trifid Cipher
 Base64 Cipher
 Fractionated Morse Cipher