An additional note to add to the good answers already here:
I've worked as a carpenter (forming, framing and finishing) some, and love sharp edges. I've never used a Dremel to sharpen; it is too small in comparison to most blades that I use. However, for a new tool or one with a badly dulled edge, I often use a "bench grinder".
For my convenience, I like to divide the sharpening process in my mind into three parts: grinding, sharpening and stropping.
Grinding is where the tool is given its general shape and the cross section of the edge is decided. A cutting edge should have as small of an angle as it can while still keeping its strength. This depends on the material to be cut; a kitchen knife used for slicing tomatoes will have a smaller angle between its two faces than an axe used for splitting firewood. This grinding and shaping of the edge can be done, as Fiasco's answer suggests, by hand with a file. This avoids the danger of over-heating. However, in most shops and factories, it is done using a powered grinder with a spinning stone like the bench grinder. If you use a bench grinder freehand, it takes patience and practice, practice, practice to learn the right angle and light pressure to hold your tool against the stone. In lighter tools like knives, chisels and plane blades, the overheating danger is much greater, and you only need to overheat the blade once to ruin it. Also, you can quickly remove too much material from the edge, and reduce the lifetime of the tool. Or you can accidentally grind too much off of the corner of a plane blade, and need to tediously grind it straight again. I repeat again from Fiasco: when grinding your tool or shaping the edge with a powered grinder, use a very light touch. However, in the case of a badly damaged edge, using only a flat stone could take a very long time to restore the edge geometry, and careful use of the bench grinder can help reduce that time. The bench grinder does not finish the edge for use; the resulting edge is uneven and has aggressive burrs that you can feel (if you touch the edge, touch softly and move your finger across the edge, not along it!) and often even see.
Sharpening is where the edge itself is prepared for use. The edge of a cutting tool is where its thickness is reduced gradually to almost nothing (almost because it still needs strength at the very edge, and where there is no material left it cannot give strength!). A general rule that I use is that if a tool makes sparks, it is too hot for sharpening. I prefer a flat stone like this:
Stones like these can be found in hardware stores all over the Americas, and probably wherever else people use knives. They are usually made of resin-bonded aluminum oxide or silicone carbide grits. You can use them with water or oil as lubricant, or dry. An important advantage of these stones over the Dremel grinder (beside not endangering the temper) is that they are wide and long enough to help you keep the angle of your tool with the stone. I often use finer stones after the one pictured, but this is good enough for many uses.
Sometimes a third step is included in sharpening: stropping or "polishing" the edge. This is traditionally done using a leather strap and very fine abrasive powder, removing the last invisible burrs left by whatever stone was used in sharpening and leaving a mirror finish on the "cheeks" of the edge.