Fritters are among the world's great foods, re-invented many times over. The basic idea is you make some batter, you cover pieces of food in the batter, and you fry it until it's delicious. There are many possible variants on this theme - savoury, sweet, spicy and so on. Almost all are remarkably tasty. None are particularly healthy.
The fritter is a slightly messier, fattier cousin of the dumpling. A fritter without anything but batter in it is just a pancake, and sometimes the lines are pretty blurry on that front.
I was raised on corn fritters, made with tinned sweetcorn stirred into a batter of wheat flour, milk and eggs, but you don't need any of those things to make good fritter batter. In fact, the single easiest way to make fritter batter is just to add water slowly to gram flour (besan, or chickpea flour) mixed with some baking powder and whatever spices you like. This is the usual way to make pakoras and onion bhajis. It makes a fairly crispy batter, largely thanks to the baking powder - it's the bubbles that do it. It also works well for a kind of vegan, gluten-free tempura, if you make it a bit runnier and go heavy on the baking powder.
You can also make good batter with many different combination of bread flour, potato starch, rice flour and corn flour - I haven't found a mix yet that doesn't work, though some flours come out dry and dense on their own. For fritters with a little bit of stretch to them, you need something that's sticky like milk, eggs or gluten - I've had good results with soy milk, ground flax seed and egg replacer, separately or together. I like to spice my batter - cumin, cinnamon, turmeric, coriander, black pepper... again, I have found so many different combinations that make delicious fritters, I don't quite know what to recommend besides experimentation. Herbs are good too - parsley is a classic addition.
Recently I have been turning assorted leftovers into fritters. Last night's big supper tasty enough, but not something you fancy eating again just as it was? Stir in enough flour to make it look like batter, with water if it's not wet enough without it, add some cumin or something if you're worried the flour might make it too bland. Then scoop dollops of the stuff into a hot pan, and fry it. Bang! Fritterised.
Like most foods, fritters are even better when deep fried, but if you prefer, you can also shallow fry them without very much oil at all, especially if you use a non-stick pan. Either way, it's probably a good idea to drain them on kitchen paper - or bread - to get some of the oil out before you serve them.
For those who like to read about specifics, perhaps with actual recipes, here are some from around Everything2 - note that most of these could be easily adapted to more specialist diets with the advice above.