As is common in science, once you get detailed about just what you are talking about, you also understand much about the thing. So: What is a day? There are really at least 4 different definitions of 'day' that we can fairly easily point to. Only two of them are still in serious scientific use. One is commonly used, kind of. And the one with the longest history is no longer in use.
What we need, in order to define a day, is something that takes 1 day to happen. The longest history for a meaning of 'day' is: "The time between maximum elevations of the sun." Almost equivalent would be time between sunsets or sunrises. Maximum elevation of the sun is a much easier and accurate measurement to make. Don't look at the sun! You also don't need to. Get yourself a stick and put it straight in to the ground (on a desk, sheet of paper, ...). Be sure that the ground is flat and the stick is vertical. Every so often through the day, mark where the shadow ends. At some point, the shadow will reach its
This notion of 'day' is affected by earthquakes, since it depends on how fast the earth is rotating. (This also makes it one of the more obscure ways of showing that the earth does rotate.)
As we got increasingly accurate mechanical time pieces, and our understanding of astronomy improved, we realized that this definition of 'day' had problems. Even if we consider the earth as rotating at an absolutely constant rate (which turns out to be a pretty good assumption), the motion of the sun through the sky is not nearly as constant. The thing is, the earth's orbit around the sun is not a perfect circle. When we're closer to the sun than usual, we move faster than usual. When we're farther away, we slow down. This shows up in the motion of the sun through the sky. The time between successive solar noons is then not very constant. The excess accumulates to several minutes each way. Consequently, we invented something called the 'mean solar day', declaring it to be exactly 24 hours long, and the hour was defined out of how clocks measured time. It's variable by something like 3 seconds per day. 3 parts in 100,000. For a long time, this was pretty good.
Contemporary with those increasingly accurate time pieces was inventing the telescope. Even before the telescope, this principle was understood. Namely, we could look to the motion of something other than the sun through the sky. The wanderers (planets) were not useful. But the fixed stars could also observed. Once we knew where north/south was -- and you can find many ancient observatories which are laid out with obvious knowledge of north/south -- we could observe the time between successive passings of the star across the north/south line (meridian). This is slightly shorter than passages of the sun, about 23.934 hours. It defines the sidereal day -- the star day. The difference between the mean solar day and the sidereal day is because of the motion of the earth around the sun. (Which means that if you have a very good watch, you can observe the earth's orbit by comparing the solar day to the sidereal day, day by day.)
With the telescope, we could be very precise about just how long it was between successive passages of a star across the meridian. Even a modest telescope (60 mm lens) can observe something that is 2 seconds of arc fairly easily. The earth rotates through 15 seconds of arc in 1 second of clock time. So with a good clock, and even modest telescope, you can measure the sidereal day to better than 0.2 seconds (2 parts per million) . Without the telescope, the limit was about 2 minutes of arc, so about 12 seconds (1 part in 100,000, not much better than mean solar day). The people who do this professionally, the International Earth Rotation Service (IERS), measure the earth's length of day (sidereal day) to 0.001 seconds, for an precision of about 1 part per hundred million. Again, this notion of day is affected by the rotation of the earth -- the passage of stars through the sky.
Our fourth, and by far most accurate, notion of day is based from atomic clocks. For them a second is some number of vibrations of some atom between particular atomic states. Last I recall, the atomic clocks are more precise than 1 part per trillion (10^12). Perhaps that's quadrillion (10^15). A day is then 86400 atomic seconds. This notion of day is completely unaffected by the earth's rotation.
The fact that the high precision day as observed by the IERS is independent from the atomic clock day is what leads to the fact that we have both UTC and UTC1 time, and the occasional 'leap second'. Some parts of society are tied to the sun and stars, still, and some go with the atomic clocks. The leap seconds bring things back together. (Most of this is due to the moon, rather than earthquakes, but that's another post.)
So 3 of our days -- the solar day, the mean solar day, and the sidereal day -- are affected by the earth's rotation. The atomic day is not. Anything that can affect the earth's rotation affects the length of day, for those 3 definitions.
The thing we need to complete our understanding is to answer: "How could an earthquake change the earth's rotation rate?" Now that we understand what a 'day' is, we see why the earth's rotation rate matters. The cryptic answer is 'conservation of angular momentum'. The visual answer is:
The skater starts out with her mass mostly away from the center of her body. As she brings her arms and legs in, she speeds up her spin. Conservation of angular momentum is the principle involved. Angular momentum is mass times how far from the line of rotation times how fast she's rotating. The arms and legs have some mass, and that isn't changing. But when she pulls her arms or legs in towards her torso, she decreases the distance. A conservation law means that the thing can't change -- in total. Since the 'how far' is decreasing, the 'how fast' has to increase. And as you see at the end, when she puts her arms out, she slows right back down.
For the Japanese earthquake, a part of the earth moved towards or away from the pole. Mass that is at the pole has zero distance from the line of rotation, so doesn't contribute to the earth's angular momentum. Something at the equator has the greatest distance from the pole. As a plate moves towards the north or south, then, it is decreasing (or increasing) its distance from the earth's line of rotation. The earth's rotation rate increases (or decreases) correspondingly.
The amount is very small, a matter of a microsecond or so for this earthquake (which means 10 parts per trillion in the length of day). Several microseconds for the Sumatran earthquake a few years ago. You see how drastic to humans something that is small to the earth can be.