"On the one hand, she was pretty, but on the other hand, she had very bad manners." (The first thing is pleasant and the second is unpleasant.)
"On the one hand, the challenger had many years of political experience, but on the other hand, he had a reputation for corruption, so I voted for the incumbent." (The first thing makes the challenger a good choice; the second makes him a bad choice.)
"On the one hand, ice cream is delicious, but on the other hand it is fattening, so I only eat it once in the while."
I see many Chinese authors trying to use this expression a lot, but it is frequently misused. One common mistake is to omit the "the," but more often I see them lead sentences with "On the one hand" or "on the other hand" even though their meanings don't contradict each other, as if the phrases were filler. It's also a little informal for a scientific paper, so I usually just take it out.(Please retain the reference in reprint: http://www.letpub.com/index.php?page=author_education_On_the_one_hand)