I found one of these in the supermarket (Ralph's market) today and brought it home to learn about it and to try it. I learned that it's an exotic fruit like a cross between a cucumber and a melon. They are also called melon shrub, tree melon, pear melon, mellowfruit, and sweet cucumber.
It's smooth and oblong (although not as much as an eggplant), yellow/off white in color with a pretty purple pattern of lines. They are apparently better if you get ones that are deeper yellow (I blew it, mine was pretty white ).
You are supposed to remove the skin and eat the fleshy inside (the seeds are good to eat as well - the inside looks like the inside of a mini cantaloupe). Pepinos are typically added to dessert plates and fruit salads. You can eat them plain but they are not very sweet (better with a little brown sugar on them).
My whole family tried it with mixed reactions. I thought it was like the inside of a large turnip (a little bland and with very similar texture.) My wife thought it was more like a melon that wasn't as sweet. I'm glad that I tried it but I probably won't get another next time I visit the store.
anthropod sent me this helpful information: A photo of a pepino can be found at http://www.foodsubs.com/Fruittroex.html#pepino. A discussion about the fruit can be found at: http://www.epicurious.com/run/fooddictionary/browse?entry_id=9540.