You can have lagged and lag free games at both 56k and broadband levels. The only difference is the type of games you will be able to play. The more involved the game is and the more information that must be traded, the better your internet connection will have to be. Thats why you haven't seen a ton of indepth fighting games and such go over 56k modems, The chance of lag has been too great. Same for sports and such. Broadband will allow a larger variety of games be online compatible plus be able to offer cooler features such as voice and stuff.
As far as what connection you need, you just need a broadband hookup, cable or DSL.
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