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Sounds a little like bitter melon. Here is a picture: http://www.flickr.com/photos/gariwerd/505268166/

There is a chinese vegetable called a Mo Qua. It does not quite fit teh description of what you are looking for, but might be related: http://chinesefood.about.com/od/vegetablesrecipes/ig/Chinese-Vegetables-Pictures/Fuzzy_Melon_Photo.htm

This article will help with growing melons: https://www.gardeningknowhow.com/edible/fruits/melons/growing-melons.htm
https://www.gardeningknowhow.com/edible/fruits/melons/hand-pollinating-melons.htm

These articles should help: https://www.gardeningknowhow.com/edible/fruits/watermelon/pick-a-watermelon.htm
https://www.gardeningknowhow.com/edible/fruits/cantaloupe/harvesting-cantaloupe.htm

This article may help: https://www.gardeningknowhow.com/edible/fruits/melons/growing-melons.htm

If we are talking about the same thing, personal melons are simply small watermelons that grow small to make it easier for a family to use up. So you will want to follow the same directions for these melons as you would with watermelons. Not only do they need to sound hollow, but the light spot needs to turn yellow before it is ripe. Here is more information: https://www.gardeningknowhow.com/edible/fruits/watermelon/pick-a-watermelon.htm

The melons will soften once picked, but will not get any sweeter if picked to early. For this reason, it is best to leave them on the vine as much as you can until as close to ripe as possible. If you must pick them before ripe, you can place them in a paper bag and in a few days they will be soft enough to eat, though you may find they are not as sweet as they could be.

The fruit can take a light frost, but should be picked after any frost has touched the plant.