The Basics
Soup dumplings (小笼包, xiao1long2bao1) are a Chinese food consisting of a skin, a solid filling, and a delicious liquid filling. These are combined, steamed in a wee basket and then served to you. There's all sorts of linguist what-not regarding whether it's considered a dumpling, a bun, or something else. This is good for the reader to note as if you walk into a random Chinese restaurant in the US and say "I want soup dumplings" you're more than likely going to end up with a bowl of dumpling soup. You must be specific.
The food of the gods
Seriously, it's been less than a year since I've had one, but my mouth waters with just the thought. Truly they are the food of the gods. Seriously. Even if you were ironically constructing some meal with mead and pomegranate and what have you, the appetizers better be soup dumplings or you've failed and offended everyone, especially the gods.
So...how do they do that?
You seal a dumpling skin with all of the fillings around room temperature, then you steam it in a wee bamboo basket.
Your inner cook is probably baffled. "Liquid? In a dumpling skin?!" your inner cook says. Meanwhile, your inner chef chortles. "Obviously, it's a gelatin" he says. When I tell your inner chef that it's actually closer to an aspic, there's much chortling and monocle adjusting while we make fun of you.
To you the instructions are more like: throw a bunch of meat, tendons, bones, etc. (chicken feet are apparently especially suited for this) into a big pot and cook them for a good long time. After much of the water has boiled off, you can cool it down to make Soup Cubes of Awesome. They're solid-ish at room temperature, so you can throw a cube into an uncooked dumpling without it splashing about. After it heats up, it turns back into a hearty broth. A mouth watering broth, hidden in a food-stuff that normally doesn't smuggle liquids.
Eating Technique
For the novice, eating a basket of soup dumplings can be a logistical challenge. They are best consumed as fresh and hot as you can possibly take them. They simply aren't as good if they've cooled down significantly below the point of injury. The liquid holds a surprising amount of heat. At the same time, the skin insulates the liquid well enough that blowing on it won't significantly reduce the temperature of the liquid. On top of that, you have fluid inside of what may be a precariously thin steamed dumpling skin. You can't handle these roughly unless you want to end up with a belly full of skin and a tabletop covered in meat and soup.
So the trick is to eat it hot, but without coating your mouth with nearly boiling liquid.
The predominant techniques I've seen tends toward either the ballsy or the conservative. On the ballsy end of thing, you pick the thing up with your chopsticks, dip the bottom in the sauce, bring it up to your mouth, nibble off the top to vent the steam, then pop it in. On the wimpy end, you get a little bit of the sauce in the bottom of a soup spoon, you transfer the dumpling to it without raising it over an inch out of the basket, poke the top with a stick, wait a bit, then finally eat it.
Regardless, approach with caution. Scrumptious, fucking awesome, caution.