For small areas, use a grout removal tool with a carbide blade. The blade is harder than the grout and will cut it away (Image 1).
For much larger areas, use a rotary tool (Image 2).
Make sure the bit is the right size for the space between the tiles, and adjust the bit to the proper depth.
Work slowly (Image 3). If the dust is a different color than the grout, you may be hitting your tile -- stay to the center to avoid damage.
Mix up a batch of new grout to match the existing color (Image 1).
Apply the grout with a margin float (Image 2).
Let it dry for about 10 minutes, then clean the area with a damp sponge (Image 3).