Depending on your browser and settings, you will either see a MD5 hashed password or the plain-text password.
Most login systems, including this forum uses sessions to keep track of its users. That would be the PHPSESSID/DarkEvilCookie cookie. The session is used as a token, if you have a valid token that matches the token value of an account in the database, you must be the owner of the account.
When you log into your account, your password will be hashed by javascript (before it is sent to the server), that is, if your browser supports it, else it will be sent in plain-text.
Then the server looks at the username and password and match them up with the database, if a user is found with the given username and password, the server will give you a session cookie (the token) and recognize you the next time you load a page.
When you log out, the token/session is destroyed.
So, to answer your first question directly; The 32 character long string that is at the passwords location in the HTTP POST packet in wireshark, is the MD5 hashed version of your password. If you want to read it/find it in plain-text you have to disable javascript when you login.