Answer: There are any number of ways to build raised beds. The main idea is to raise the level of the planting bed above that of the surrounding grade. You may want to do this to improve the drainage conditions or to provide a better soil for the plants to grow in or both. In any case, the soil in the raised bed is piled on top of the existing soil -- there is no "bottom" to the raised bed. For this reason it is always a good idea to loosen the existing soil and if your bed is not too high, to improve the underlaying soil as well to accommodate any plant roots that extend into it.
You will need to raise the bed at least a few inches or perhaps as much as a foot. The soil you put in the raised beds can be simply the existing soil mixed with amendments, or it can be imported soil, or it can be a mixture. Working the soil will "fluff" it up so that alone will provide some of the height. You should run some basic soil tests to see what if any amendments would need to be added to optimize your soil. Your County Extension should be able to help you with the tests and interpreting the results. In general though, the addition of copious amounts of organic matter such as compost, rotted leaves, and aged manure and bedding will help the most.
You may edge the beds, or not. Some people prefer the tidier look of hard edges such as wood or cinder block, others are happy with unreinforced sheer, steep sides which simply shed water. To increase the available square footage for planting, gently sloped sides are better.
Make the beds of a size and shape that you can reach all the way to the middle comfortably -- one of the beauties of raised beds is that the soil is never compacted by being walked on. You might also take into consideration any watering system that you use so that the beds fit it well. Other than that, it is basically a matter of personal preference and perhaps whatever materials you have on hand or easily available.
When you plant, since the space is relatively small, you will probably use some version of the square foot gardening technique or a similar intensive method with successive plantings. You might want to read up on that a bit. One book I particularly like is the classic "Square Foot Gardening" by Mel Bartholomew, ISBN 0-87857-340-2.
Enjoy those raised beds!
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