So we have two problems here, we want a nice evenly chopped onion -- that everyone else seems to do so easily -- and we don't want to cry in the process.
First, let's address those tears. You are less likely to need tissues if you use a red onion rather than a white or brown one, but obviously this depends on the recipe -- sometimes only a big Spanish onion will do!
Now here comes the science bit... When you cut into an onion a form of gas is released which is the reason for the tears. There are many methods bandied around, but the sure fire one that works is to breathe only through your mouth and not your nose. If you breathe through your nose then you are inhaling that gas which goes straight up the nose and aggravates your tear ducts.
Of course, you could just remember to breathe through your mouth only, but whilst concentrating on chopping you could try holding a wooden spoon handle (or similar) in your mouth; keeping your mouth open and forcing you to breathe that way only. Personally, the thought of drool on my utensils is usually enough to motivate me to just remember to essentially cut off my nose from the breathing process when chopping, but I'll leave that to you!
As for the chopping, the best way is to cut off the top bit of the onion (the pointy bit), peel, and cut the onion in half through the middle of the root.
Lay an onion half face down on your chopping board. Make cuts through and down the onion, but do not cut into the root -- keep your cuts maybe 1/2 inch away from the root. If you want slices then all you need to do now is to chop off the root and you are left with perfect slices and no mess.
For a chopped onion, turn your onion 180 degrees and chop to create your little onion pieces, finishing at the root end.
Essentially with both of these methods you are using the root as a holder for the rest of the onion as you chop.