You can only summon other players while in Body Form. You must be in a level of your own game world where the Boss Demon is still alive. Once you have defeated the Boss Demon you can still go back and explore the level, but you can no longer summon other players to aid you. When the Boss Demon is defeated the co-op session is over. The Blue Phantoms are returned to their own world, where they are revived in Body Form themselves. You can then move on to the next level, where you can once again summon Blue Phantoms.
When playing in Soul Form you will have only half of your life bar, but you will generally be able to do more damage in combat. You cannot summon other players. Other players' Blue Eye Stones won't even appear in your world, so long as you are in Soul Form. You can place your own Blue Eye Stone almost anywhere in a level. You will want to set it in a nice open location. If you hide it, no one will summon you. You can either wait around to be summoned or continue on through the level. A notification message will eventually appear, saying "You are being summoned to another world," and then you will spawn in a Host player's game as a Blue Phantom. You will then play the game as you normally would, except you cannot pick up items unless the summoning Body Form player drops them for you. Don't count on it.
Once you defeat the boss you will return to your own world, now in Body Form. You will most likely have a large number of souls for your efforts which you should spend immediately. What really matters is the practical game experience you've gained. Now you have been through the level, and you will have seen enemy weaknesses, what techniques work best against the boss, and the locations of items and incredibly important shortcuts. Unfortunately, you will still have to beat the boss in your own world. None of the level progress from your Blue Phantom adventures will carry over to your game. Now, of course, you are in Body Form, so you can summon your own Blue Phantoms to aid you. Once a boss is defeated in a world it will not respawn until a new-game-plus has been started.
Death is a harsh mistress in Demon's Souls. If the Host Player dies the co-op session is over. The Host Player will respawn in Soul Form in their own world. Any souls they may have gained in the session will be left in a soul-drenched bloodstain near the location of their demise. Blue Phantoms are returned to their own world in Soul Form with whatever souls they had intact. If a Blue Phantom is defeated, they simply reappear in their own world, once again in Soul Form.
There is no Friends List invite system or in-game lobby. Co-op play is based upon your silent interaction with strangers. If you want to team up with a friend, you'll have to make co-op arrangements about when and where one of you will drop a Blue Eye Stone through some other means. And then you'll have to be within level range of each other, and it will have to be in an area where the host player hasn't defeated the boss. Inventive players will figure out how to play cooperatively, with voice chat, and with their friends. (We used Xbox LIVE. Blasphemy, I know.)