Hang on a sec, folks.
There are a lot of different kinds of kimchi, which are not at all made the same ways.
The kind most people are familiar with, or most familiar with, is traditionally made primarily from Napa cabbage, seasoned with salt, garlic, chilies (do NOT substitute cayenne for the Korean-style chilies, incidentally), and quite often seafood. The seafood, commonly a type of anchovy (used raw and split open, guts left in), contains a number of factors that produce a controlled fermentation extremely similar to what makes sauerkraut work. These days, the seafood is normally added only as a flavoring factor; "kimchi juice" is used to start the reaction in a controlled and safe fashion.
This type of kimchi should either be eaten relatively soon after making it, or else allowed to ferment well in a cool place, roughly 45-50 degrees F I am told, for at least a month and often longer. The two results are very different, and used differently. Perfect kimchi chigae, for example, which is a stew made from kimchi and a range of possible other things, should be made from the ripe, old stuff. For eating as a side dish, it's a matter of preference and taste which kind you prefer. In the old days, the difference was simply when you made it: if you made it in warm weather, you ate it soon; if you made it at the end of the season, you packed it in crocks and buried them.
Other kinds of kimchi are made from other vegetables, combinations of vegetables, and a range of flavorings. Some are blazing hot-spicy, and others (particularly from the northern regions) are very mild, more akin to Japanese tsukemono than what you might think of as kimchi.
To make a pretty standard non-cabbage kimchi, try cutting daikon in generous 1" cubes. Add sliced garlic, dried chilies, and lots of salt, and place in a pickling jar with a couple of inches of headroom. Cover with a couple layers of cheesecloth and wait, leaving it someplace medium-cool, like say 60-65 degrees F. After several hours, stir it up a bit and then pack it down in the jar gently. At some point the liquid extracted is going to go over the top of the daikon cubes; if they start floating, use a very clean small jar lid or the like to hold them under. Now put the thing in the fridge, or if you have it a cool wine cellar or the like, at about 45-50 degrees F. Let process for several days to a week, then eat. The daikon should be crunchy, acid in the way sauerkraut is acid, and quite spicy (though it is sometimes made quite mild). Now that you have the concept, feel free to vary every part of it: the only part that really matters is that if you want to let it "percolate" several days, you need the liquid to be over the top of the vegetable; if need be, add salt water, and feel free to add a small splash of mild rice vinegar as well.
There are many other kinds of kimchi made by more specific and complex methods, but this is a sort of baseline for a great many of them. You can do the same thing with fresh kimchi pickling, in which essentially you do the same process but don't bother waiting for the liquid to rise: you serve after a couple-three hours. This works well with cucumber slices, for example, and seasonings often include citrus zest, ginger, and the like as well as the usual garlic and chili (which may be minimal or even eliminated).
To make cabbage kimchi, just search the web for recipes: there are a zillion, and they're all quite similar. Just be sure that if you want fermented kimchi, you do not make the mistake of confusing kimchi juice or "Korean fish sauce" with the Southeast Asian version of the same: the two can be interchanged, but I am told that the normally-available Southeast Asian fish sauces are processed such that they will not produce the desired reaction reliably. Any actual seafood used in kimchi-making should be impeccably fresh and clean, and should be used exclusively for flavoring, not as a fermentation catalyst -- I recommend the many excellent Japanese salted-dried seafood products for use this way. Normally cabbage kimchi does not sit under water; it does generate a good deal of liquid, but it's packed densely and doesn't really float much.
