The right way ("right" being defined as fast & easy) is to cut the flesh away from the core, not to try to cut the core out of the pepper. Basically you want to chop off the outsides the same way you take the outside off a pineapple. Also never detach the stem part of the top from the core, you want it to hold the core together so you don't make a mess.
In detail : stand the bell pepper up vertically ; put the most stable end down, whatever that is. Your non-knife hand should be stabilizing the pepper by placing a finger tip at the top. Make vertical cuts downward with your knife, following the shape of the outside of the pepper, to cut off just the edible pepper walls. The knife should go right along the inside of the pepper walls so that the veins are left attached to the core. Don't try to cut too much in one slice or you will cut into the core or the veins - just take off the outside. It will take 3-8 cuts to take off all the outside depending on how irregular the shape of the pepper is. Your non-knife hand should now be holding the core with all the seeds and veins attached, which you throw away. When you do it right (very easy with practice) you will be left with 6-8 slabs of edible pepper exterior with almost no veins or seeds, completely ready for use.
For example, all of these people are 100% retarded. retard 1 , retard 2 Like hey, I know nothing about anything and my skills suck and my brain is broken, let me make videos telling people how to do things that is totally the wrong way. The blind leading the deaf or something. Yay interwebs.
(I mistakenly had this guy in the retard list, but actually he just shows the wrong way first and then the right way, not retard )
This guy at video.about.com roughly does it right, but his knife skills suck donkey balls. Maybe this is how you will fare since you probably suck ass with a knife too. Go practice dicing onions.
Okay I made my own video . I only had one bell at home so I knew it had to get it in one take. The pressure was on so I went extra slow and careful.
BTW I don't mean to rag on the amateur home chef with shitty skills. This is for you to learn. The thing that pisses me off are these people who presume to be teachers and just suck so bad. How dare you think that you are qualified to teach anyone anything when you obviously haven't studied or put any effort into this subject at all !? I have great respect for the people out there who actually have skills, and there are plenty of them, and it's disgusting for someone to put themselves on that level when they don't deserve it.