How do you break encryption on a message?
As mentioned this is a very broad question, so please forgive me for not going into any great depth on each point.

There are a few ways to beat encryption. One way is to attack the actual math of the cryptography: for PGP that would involve cracking RSA, which would involve finding a way to solve the discrete log problem. This is the hardest method, but also the most complete. The second way is to attack particular implementations: a specific implementation might leak key information through how long encryption and decryption operations take, might leave secret data in memory for an attacker to find, etc. This is often combined with hacking the target's computer, possibly getting access to the plaintext and circumventing the encryption. "Extraordinary rendition" (kidnapping) and "enhanced interrogation" (torture) are other popular techniques for the US intelligence agencies to break encryption. Just torture the target until they give you a valid key. 

There generally isn't a hash of a key transmitted. I'm not sure where you got this idea.

Breaking passwords is easy because people chose bad passwords. It's done via trying the most common passwords and seeing which ones hash to the matched hash. Encrypting things based directly on a password would be a terrible idea, key derivation algorithms are used to try to make breaking passwords harder, but they still rely on the user choosing a good password. Also note that hashes and encryption are very different things. They're both part of cryptography, but they serve very different purposes.

In general, there is no easy way to decrypt an encrypted message, so long as the encryption was done with a good algorithm and good implementation. If there IS an easy way to decrypt an encrypted message then the encryption system used is by definition broken.