Unlimited bandwidth, in most aspects, is not possible. A server can only handle so much in/out bandwidth. A company offering "unlimited" bandwidth will still limit you to a certain acceptible usage amount and will suspend you if you go over it.
A fully managed vps/sever means that the company which you buy the server from will setup the server, test it, keep it updated, monitor it, administor it, and basically keep it functioning properly.
If you have problems, they will usually fix them for you.
Semi-managed vps/servers -> they would set up the server for you and monitor it. They will fix any hardware issue's but it's up to you to administor the server, secure it and keep it functioning.
If you run into problems, they may sometimes charge you, if you need them to fix it for you.