If the admin really wants to see your password, then he will be able to do so. He just has to grab it when you send it to the server.
If you want to know what password-dependent data is stored on the server side, then that's another question. In general, you cannot know how they do things, but sometimes you get clues. Basically, the reasonable thing to store is a "password verification token" which is a hashed version of the password, using a proper password hashing function. If they do that, then they cannot retrieve the password and send it back to you. Thus, if there is a "password modification screen" that can show you your password, or if they send back your password to you by email when you click on the "I forgot my password" button, then you know that what they store is unreasonable. Plain Text Offenders is a rage/shame Web site that gathers reports of such occurrences, mostly for being able to point and mock these poorly developed sites.
What you cannot really know from the outside is whether, when they store only hashed passwords, they use a proper password hashing function, or something else. You usually learn how poorly they did things when they get breached in and a list of hashed passwords is extracted: if the passwords get broken at a rate of dozens per second, then the hashing function was quite bad...
Password reuse is too dangerous. You should use a new, specific password for each site -- that way, any ineptitude on their side will be contained to that specific site. To remember all these passwords without having an enormously efficient brain, you may want to use a password manager software, e.g. KeePass.