There are some simple answers to your questions, and some (historic) controversy. I'll answer the simple elements first (but backwards by your ordering of the questions), and then describe the nature of the controversy.
- What is absolute motion?
Absolute motion is motion that is referenced (compared to) a feature that you can say is standing still. For example, how do you know the speed of a car? You compare it to something that is standing still.
- What is relative motion?
Relative motion is motion that is referenced (compared to) something that may be moving. For example, when driving, you may find that the car next to you is edging ahead of you. You can say that it is moving 5 miles per hour faster than you - that is its relative speed. (Note, it also has an "absolute" speed, say 65 miles per hour, while you have an "absolute" speed of 60 miles per hour. Here I use "absolute" in the sense of comparing speed to the surface of the Earth, not a universal absolute.)
This brings us to the definition of a reference frame, which is critical for understanding the later arguments. A reference frame is a perspective from which one can begin to measuring things. It is a defined zero point from which you can measure distance, speed etc. So, in the example above, you have a reference point, yourself in the car, and you see the other car moving ahead of you at 5 miles an hour, and you also see the ground moving past you at 60 miles an hour. The driver of the car also has a reference point, himself, and sees you as moving backwards at 5 miles an hour, and the ground moving past at 65 miles an hour. Someone sitting on the ground also has a reference point, seeing two cars moving away at 60 and 65 miles an hour respectively.
- How do we know that indeed the earth moves?
We compare the Earth to other things. We can compare it to the sun, and therefore know that it is both moving around itself (rotating) and around the sun (revolving). We compare the sun to the milky way, noting that the entire solar system is orbiting around the center of our galaxy. We compare the galaxy to other galaxies, and note that our galaxy is drifting towards our local great attractor. And (this is where the controversy of the other answers comes into play) we can compare our speed to the equivalent motion of an object at rest with the Cosmic Microwave Background Radiation (called CMBR for short), and can even deduce how fast the local great attractor is moving compared to that.
So what is the controversy? The controversy is around the question, "Is there a universal reference frame?" In other words, is there such thing as a point in the universe that we can say is the zero point that we measure all positions and velocities from?
Historically, this was fought out between Einstein and Lorentz. Before the Michelson-Morley experiments, it was thought that light traveled through a ghostly material called an "aether" that wasn't moving, which could be considered the universal reference frame. The Michelson-Morley experiments were set up to determine the Earth's relative motion through the aether, but it didn't find any. Lorentz created a mathematical method that could account for this, but it was missing a physical basis. Then along came Einstein, who replaced the assumption of an aether with the assumption that light travels at a constant speed. From this he developed the now famous Special Theory of Relativity (although, that is the name it has subsequently come to be called). This was a little too far for Lorentz, and he developed a counter theory, which slowly transformed into a very similar theory to Einstein's, with one exception - Lorentz believed that there was an, as yet undetectable, zero reference frame for the universe. Here is what he said in 1910
Provided that there is an aether, then under all systems x, y, z, t, one is preferred by the fact, that the coordinate axes as well as the clocks are resting in the aether. If one connects with this the idea (which I would abandon only reluctantly) that space and time are completely different things, and that there is a "true time" (simultaneity thus would be independent of the location, in agreement with the circumstance that we can have the idea of infinitely great velocities), then it can be easily seen that this true time should be indicated by clocks at rest in the aether. However, if the relativity principle had general validity in nature, one wouldn't be in the position to determine, whether the reference system just used is the preferred one. Then one comes to the same results, as if one (following Einstein and Minkowski) deny the existence of the aether and of true time, and to see all reference systems as equally valid. Which of these two ways of thinking one is following, can surely be left to the individual.
Einstein maintained that not only was a universal reference frame undetected, it was unnecessary, and Occam's razor should rule (i.e. that if something is unnecessary for the theory to work, then it shouldn't be included in the theory). It is interesting that both theories actually predict the same things, but in the end, Einstein won. Most physicists these days would state that a universal reference frame does not exist.
Since that time, two key things have happened that impact this. First is the discovery of the expansion of the universe. This means that, even if we were to make an arbitrary universal zero position and time, other reference frames that start out at stationary compared to this reference frame will start to move away, even though any object there has undergone no acceleration. This greatly complicates any attempt to set a universal reference frame, an makes it much easier to consider that none exist. Tick one for Einstein.
The second thing was the discovery of the CMBR. This is the relic of a change in the entire universe at around 380,000 years after the big bang. At that point, the universe went from opaque (where light couldn't travel very far without encountering some matter to interact) to transparent (where some light could travel consistently without interruption until someone like you measures it). Two interesting (and relevant) features were found about this CMBR - it comes from every direction (consequently it is universal, i.e. can be seen from anywhere), and it is very consistent in intensity from every direction (termed isotropic). This means that anyone, anywhere, can measure their speed relative to the local knowledge of the universe. Furthermore, this effective reference frame can be used to measure velocity irrespective of the expansion of the universe, if one considers certain interpretations about comoving reference frames. Although this is very different to a universal reference frame of an aether, it is certainly, in one sense, a universal reference frame. Tick one to Lorentz.
So what does this mean? Does it mean that Einstein was wrong? No. Einstein was concerned with the physics of light and electrodynamics, of the relationship between time and position and velocity. The basis of Einstein's argument was that, no matter how fast you go relative to any other object, the physics of what you measure locally will be the same. The original concept of a universal frame implied otherwise, and Einstein has been proven experimentally correct over that original concept.
However, the concept of using the CMBR to define a universal reference frame is a slightly different idea than the original concept (and more in line with Lorentz's final theory). It does not imply any change in the local physics due to relative speed. But it is, as suggested by Rob Jeffries above, a useful reference frame. I would argue it is a special reference frame, in a manner that science has created special zero points throughout history (e.g. sea level, 0 degrees latitude, binding energy etc). None of these zero points change the physics of what is measured locally, but they make it much easier to compare measurements at different locations etc. In the same way, the CMBR reference frame makes it easier to answer a range of questions about non-local physics. For example, what is the age of the universe? (The age is maximum at the CMBR reference frame.)
So, in conclusion, when we see some arguments between physicists about whether the CMBR is an absolute or universal reference frame, it is usually because one side or the other is interpreting the term in the same manner as the original concept, with all its incorrect implications, rather than the lesser concept of a natural definer of the local speed to a universal feature of the universe.