There's a lot to speculate when it comes to the specific act of summoning, but referring to the Season 1 trailer, the specific tidbits revealed by champion bios (as Sir Regendorf stated with Ezreal), and champion in-game quotes, you can piece together the process.
Summoners call the champions to the fields of justice (definitely physically, imo) from a location in the League, and use their magic to communicate with them. They are not necessarily in control of them, as this might reduce the champion's expertise, but it is simply said that their 'minds are linked'. This could vary from complete control to nearly nothing (as I'm sure would be the case when it comes to someone like Swain, who would likely refuse any entrance to his mind). I also imagine they would need greater control over a champion like Cho'Gath or Renekton, simply to ensure they continue to fight for the summoners' objectives.
While the game itself doesn't leave any room for champions to disobey summoners' requests, I imagine if it were a real-life situation you'd have it happen every now and then. For instance, "I'm not going to fight all five of them under their tower, you dumbass!"
From there on, including all the details of fighting, dying, items, well... it's up to your imagination. :P