This is a classic question - and a wonderful one!
The short version of the answer is this: cutting the onion releases certain substances that, when they come into contact with water, produce sulphuric acid. And since your eyes and nasal mucosa are moist, this reaction happens there. Understandably, dropping an acid (even this mild one) into you eyes is enough to make them water in response, and so you 'cry'. The running nose is partly from the same reaction happening on your nasal mucosa, but also partly because tears are drained by a duct that feeds into the nose - any increased tearing makes your nose 'run'.
The longer and even more interesting version is available here. Definitely worth a read.