Questions about how fast the earth--or anything, for that matter--is moving are incomplete unless they also ask, "Compared to what?" Without a frame of reference, questions about motion cannot be completely answered.
Consider the movement of the earth's surface with respect to the planet's center. The earth rotates once every 23 hours, 56 minutes and 4.09053 seconds, called the sidereal period, and its circumference is roughly 40,075 kilometers. Thus, the surface of the earth at the equator moves at a speed of 460 meters per second--or roughly 1,000 miles per hour.
As schoolchildren, we learn that the earth is moving about our sun in a very nearly circular orbit. It covers this route at a speed of nearly 30 kilometers per second, or 67,000 miles per hour. In addition, our solar system--Earth and all--whirls around the center of our galaxy at some 220 kilometers per second, or 490,000 miles per hour. As we consider increasingly large size scales, the speeds involved become absolutely huge!
The galaxies in our neighborhood are also rushing at a speed of nearly 1,000 kilometers per second towards a structure called the Great Attractor, a region of space roughly 150 million light-years (one light year is about six trillion miles) away from us. This Great Attractor, having a mass 100 quadrillion times greater than our sun and span of 500 million light-years, is made of both the visible matter that we can see along with the so-called dark matter that we cannot see.
Each of the motions described above were given relative to some structure. Our motion about our sun was described relative to our sun, while the motion of our local group of galaxies was described as toward the Great Attractor. The question arises: Is there some universal frame of reference relative to which we can define the motions of all things? The answer may have been provided by the Cosmic Background Explorer (COBE) satellite.
In 1989, the COBE satellite was placed in orbit about the earth (again, the earth is the frame of reference!) to measure the long-diluted radiation echo of the birth of our universe. This radiation, which remains from the immensely hot and dense primordial fireball that was our early universe, is known as the cosmic microwave background radiation (CBR). The CBR presently pervades all of space. It is the equivalent of the entire universe "glowing with heat."
One of COBE's discoveries was that the earth was moving with respect to this CBR with a well-defined speed and direction. Because the CBR permeates all space, we can finally answer the original question fully, using the CBR as the frame of reference.
The earth is moving with respect to the CBR at a speed of 390 kilometers per second. We can also specify the direction relative to the CBR. It is more fun, though, to look up into the night sky and find the constellation known as Leo (the Lion). The earth is moving toward Leo at the dizzying speed of 390 kilometers per second. It is fortunate that we won't hit anything out there during any of our lifetimes!