Raspberries come in 4 basic colors, and two bearing types. Red raspberries are the most common, and derive from something ideaus. Whether or not the north american and the eurpoean sources are related species or different species is debated, but they will interbreed. Yellow raspberries are a variant of red raspberries. Often times they are closer to being orange than yellow.
Black raspberries are a different species, originating in North America. They tend to be more susceptible to disease and less tolerant of extremes in growing conditions. They also are often smaller plants than red and yellow raspberries. While red raspberries often do well with an upright trellis system, black raspberries are often trained to a lower, more spreading trellis. Purple raspberries are a hybrid between black and red, but generally grown like a black raspberries.
All raspberries are most often grown in rows. The rows are spaced far enough apart to accomodate whatever sort of machinery will be used near them. For home gardens this is usually going to be a lawn mower. I like to space mine twice the width of my mower apart in rows, or something in the neighborhood of 10 feet. This gives enough space for the raspberries to overhang somewhat without causing me a lot of blood loss from brushing again thorns while I mow.
Mulching the plants helps them a lot, as it keeps their relatively shallow roots moist, keeps down competing weeds, and the decaying mulch increasing soil fertility. Do not mulch between the rows, as this encourages suckering in the walkways between the rows.
Within the rows, two feet is a good distance to set the plants. Ultimately the plants will etnd to grow into a solid hedgerow. If you set the plants 3 to 4 feet apart, it will take longer for the row to come into full production. Closer than 2 feet brings the plants into a solid row quicker, but requires more money to buy plants for a given length of row. Eventually crowns will need to be thinned out.
When planting new plants set the crown close to the same depth it was originally. Planting too deeply encourages crown rot.