Short version: Earth's average orbital speed is about 30 kilometers per second. In other units, that's about 19 miles per second, or 67,000 miles per hour, or 110,000 kilometers per hour (110 million meters per hour).
In more detail:
Let's calculate that. First of all we know that in general, the distance you travel equals the speed at which you travel multiplied by the time (duration) of travel. If we reverse that, we get that the average speed is equal to the distance traveled over the time taken.
We also know that the time it takes for the Earth to go once around the Sun is one year. So, in order to know the speed, we just have to figure out the distance traveled by the Earth when it goes once around the Sun. To do that we will assume that the orbit of the Earth is circular (which is not exactly right, it is more like an ellipse, but for our purpose a circle is close enough). So the distance traveled in one year is just the circumference of the circle. (Remember, the circumference of a circle is equal to 2×π×radius.)
The average distance from the Earth to the Sun is about 149,600,000 km. (Astronomers call this an astronomical unit, or AU for short.) Therefore, in one year, the Earth travels a distance of 2×π×(149,600,000 km). This means that the speed is about:
speed = 2×π×(149,600,000 km)/(1 year)
and if we convert that to more meaningful units (knowing that there are, on average, about 365.25 days in a year, and 24 hours per day) we get:
speed = 107,000 km/h (or, if you prefer, 67,000 miles per hour)
So the Earth moves at about 110,000 km/h around the Sun (which is about one thousand times faster than the typical speed of a car on a highway!)
Thanks for your explanation, but I was hoping for an explanation a little more precise, since I already knew the one you gave.
In the case of your question about the speed of the Earth around the Sun, there isn't really a more 'precise' answer. The only approximation I did in the calculation I sent you is assuming that the orbit of the Earth is circular. This is in fact a very good approximation. One of Kepler's laws describing planetary motions states that all orbits are ellipses. This is the case for Earth's orbit. But not all ellipses come in the same shape. They are described by their 'eccentricity', which tells us how flattened they are. The eccentricity of an ellipse is a number that varies between 0 and 1, 0 being a perfect circle, and close to 1 being a very flattened ellipse. It turns out that the orbit of the Earth right now has an eccentricity of about 0.017. This means it is almost a circle, making our approximation valid. So under the one approximation that was made, the calculation couldn't really be more 'precise'. And as for the average Earth-Sun distance, the true value changes slightly over time due to gravitational perturbations from the other planets, so there really isn't much point in using a more precise value than the one given above.
Now if you want to calculate the speed of the Earth on its orbit without assuming it is a circle, it is another ball game! First of all, I cannot give you a precise answer, because the speed of the Earth changes all the time as the Earth moves around the Sun. This is because Kepler's second law says that on its orbit, a planet will sweep equal areas in equal amounts of time. This means that when the Earth is closer to the Sun (which happens in early January, about two weeks after the northern winter solstice) it's moving faster than when it is farther away. (For more information on how the Earth's orbital speed varies over the course of a year, please see this answer.) Unless you specified a certain date, this means I cannot give you a precise value for the speed of the Earth assuming its orbit is an ellipse. We are better off to stick with the first number we got - the average speed.