The answers at What is "angle of view" in photography? should help. In short, the "equivalent" gives you a way to compare the angle of view of these lenses, by putting them all in the terms of a common format.

The format usually used as the standard is that of normal 35mm film — called "135" because that's the standard film cartridge format of that size. This format is also used by "full-frame" DSLRs. However, that's (currently and for the forseeable future) the realm of high-end cameras, usually well above $2000 for the body with no lens. Most DSLRs use a smaller format called "APS-C", which is about half the area. There's actually a number of slightly different sizes that go by this name.

For these smaller formats, you can get the "equivalent" field of view by multiplying by the "crop factor". For Sony, Nikon, and Pentax, the value is 1.5x. For Canon, it's 1.6x.

To put that in concrete terms, with the Nikon D5000, a 15-390mm lens would have that same 22.5-585 equivalent angle of view as your example. (Because 15 x 1.5 = 22.5, and 390 x 1.5 = 585.)

The lenses in compact cameras are often specified only in terms of their equivalents, because a) bigger numbers are more impressive and b) there's a dizzying array of sensor sizes in compact cameras, so using some standard makes sense. A typical sensor size is called 1/2.3", and that has a crop factor of about 5.6 — so a compact camera advertising a 22.5-585mm lens may really have a 4-105mm lens.

But don't be too swayed by the impressiveness of that gigantic zoom range. It comes at a cost — one of course being that it's easier to do with the smaller sensor coverage, and those pinky-finger-nail-sized sensors are at a disadvantage, especially in low light. But that's not all, and the lens will certainly also have significant compromises in sharpness, chromatic aberration, distortion, bokeh, and every other aspect of image quality. It still may be very capable of good results, but you should be aware of where the gear you are using makes compromises. See What does 'how much zoom' mean? for more on this.

Finally, remember that zooming is functionally equivalent to cropping — that's why the term is "crop factor" and why this equivalence works out in the first place. That means that if you have an image shot at a "300mm equivalent" focal length (for example, with a 200mm lens on a typical DSLR), and you take 25% off each edge, the resulting cropped image has a field of view like a 600mm lens. Because of the larger sensor and potentially better lens, this cropped image probably will look even better than the "full" image from the point-and-shoot with the 585mm-equivalent lens. (A decade ago, pixelation might have been a concern, but now even entry-level DSLRs have more megapixels than needed for reasonably-sized prints even after such a crop.)