There are several common file types used with regards to email.
EML files are the most common type of email files. An EML file contains the raw message content. Most email software can export and import EML files simply, because that is how the messages are stored internally. For instance, in Mozilla Thunderbird, if you perform save as on a message, it will save it as an EML file.
Because the EML file format is the raw message content, all the Internet standards relating to message format such as RFC 5322 apply to EML files as well.
MBOX files are 'mailbox' files. They are quite widely used in email software to store the contents of mail folders. An MBOX file is essentially a concatenation of all the email messages (in raw format).
RFC 4155 describes the format of MBOX files.
MSG files are the Microsoft proprietory format for saving files. Microsoft Outlook will save files as MSG files instead of the standard EML file format. Generally, of email clients, only Microsoft Outlook can read and write MSG files natively. There are utilities available for converting MSG files to EML files and vice versa. However, because Outlook converts messages into its own internal format for storage in its PST files, the MSG file often cannot be perfectly converted back into the original raw message, unless email clients which store messages in the original EML format (or in MBOX files)