Getting the encryption method and key from the encrypted data and the raw data I have some pairs of plaintext and ciphertext data, from which I need to be able to decrypt the other passwords stored in the database.

The password field typically contains something like


ENC:LCznyFywN+3J+AD330rqHg== 
ENC:D3H2ODVCAzkzQzRr8ZbJdw==


The raw data for the first is 1234, and the second 62369. ENC: denotes that it is an encrypted field, and the == at the end is always the same (no matter what the password). 

If this can be done, then any prods in the right direction would be greatly received.
There is no general method for that. Cryptography is about making it impossible in practice.

In the particular examples, it looks like "encrypted data" is encoded in Base64, as 22 characters, representing 128 arbitrary bits. From that it can't be deduced much about the scheme used.