Whoa!! You’re using an old version of Blender. Yes, I know it sucks that nice looking tutorials don’t show the current Blender interface, but that’s progress for ya… You can of course always load the Blender version the tut was made in…
Currently the effects panel panel can be found with F7 (Object ) and then clicking the Physics button. There, you’ find your Particles panel with a “New”-button Mind you, buttons within this panel have also largely changed so good luck finding the buttons you want…
Of course, in the next iteration this will change again and the Physics button will be split over a Physics and a Particle button (in view of the MAJOR rewrite of particle code) .