That largely depends on the Operating System/Trading Software etc etc that you're using. Coupled with the fact you need you machine to be responsive so as not to have any delays when monitoring charts opening windows etc.
I personally am running a P4 3.2ghz which handles what I need it to do without even getting out of second gear so to speak. Most systems available today (new) usually have nothing less that a P4/ Celeron processor running at 2.2ghz upwards which is perfectly suitable. There's plenty of bargains to be had out there.
Probably more importantly is the speed of your internet connection, you really need to be on ADSL/Cable to get the real time data through quick enough to be relevant.
Feel free to pm me if you want some advice.
Regards,
Graham