If all I could afford was one electric appliance, I would have to pick the Electric Wok. It can do many different things from frying to sautéing to steaming and packs quite a punch. The sloped sides are better than an electric frying pans because it uses less oil and you can place food on the sides to regulate cooking speed. You can find a cheap one at a thrift store, but make sure it has all the pieces.
Additional Equipment:
Small Refrigerator - If you don't have a kitchen, you DO need a small refrigerator. It just makes life a little easier. If you can't afford a small refrigerator, you will have to go grocery shopping more often. You will also have to be good about portions as you don't have a place to keep them fresh.
Toaster Oven - This would be the third thing I would buy. It has many, many uses. It can heat frozen pizzas, make toast, bake cookies, roast vegetables. The possibilities are endless.
Rice Cooker - There are a surprising amount of uses for rice cookers. Sure most of them do include rice, but rice is a staple.
Crock Pot - These are tried and true. Slow cookers are amazing if you have at least 4 hours between when you prepare and when you eat. It also bakes. That makes it a versatile friend.
Spice/Coffee Grinder - In some ways, I would put this second. When you want great tasting food, freshly ground spices make all the difference. It can also save money and time in the long run. Whip up a quick batch of Cajun Seasoning, and you have a spicy mixture you can throw on lots of things. Oh...the prospects!
Microwave - Of course this is an obvious suggestion. And it is a good tool. But if what you want it fresh and tasty, the microwave isn't your best option. It's perfect for reheating, melting, boiling, popping, and that's pretty much it.
Assorted Equipment - There are various other pieces of electrical equipment, but they all over the place in terms of function, ease of use, and versatility. I am not a huge fan of the George Foreman Grill or his other products. Too clucky and not enough variety in what can be made. Sandwich presses are in the same category. I found a rotisserie at Community Thrift, but the heating element is on the bottom right where the fat drips, and it seems to be a recipe for disaster. That and after unpacking everything it didn't come with a plug.
Knives - These are essential. If you only can afford one knife, I suggest a regular chef's knife. You can use it for pretty much everything. Sure, it is larger to handle than a pairing knife, but it makes everything else easier.
Cutting Board - If you have a knife, you'll need a cutting board. Not only is it hand for chopping, it makes a great tray for presenting food. I prefer bamboo cutting boards because they look nice and are durable. Plastic are the least desirable.
Colander - If you boil water for pasta, you will want something you can drain it in to. You can do drain liquid holding the lid slightly closed, but it is bound to end in a mess. It also could result in electrocution or a ruined piece of equipment if water or liquid spills on the heating element or plug. A colander reduces that possibility. And...it is perfect for washing food in if you have a shared bathroom. Who wants to put their fruits and vegetable directly in that sink? A colander provides a useful protective barrier.
Mixing Bowls - One really large bowl is perfect. Two is even better. Then, smaller bowls. Think of it this way, you can always mix a small amount in a large bowl, but you can't mix a large amount in a small bowl. Also...if all you can get is one bowl, try and NOT get a metal one. If you mix vinegary or citrusy things in it, the sauce retains a tinny flavor.
Wooden Spoon - These are preferred over metal only because they work on everything. You can't use metal on anything that is non-stick. Wood also doesn't heat up to the point of burning if you leave it sit in the electric wok.
Other Utensils - There are a bunch of other utensils that are handy including a spatula (both kinds), a whisk, a slotted spoon, a strainer, a cheese grater, and a peeler. They certainly make cooking easier. But if you are on a limited budget, the knife, bowl, and spoon are the top three things you will need.