Weight and mass are two different things:
The mass of an object is an intrinsic property of the object itself. Mass does not have a direction. Mass is often expressed in kilograms. A person's body has the same mass whether that person is standing on the earth, standing on the moon, or falling off a roof.
The weight of an object depends on how it is moving, and weight has a direction. An object in free-fall is weightless. An object that is not in free-fall has weight. The amount of weight is has depends on how much the object is deviating from free-fall, and the direction of its weight depends on the direction in which it is deviating from free-fall. Weight is often expressed in pounds or Newtons. The same person's body has one weight when standing on the earth, a different weight when standing on the moon, and no weight at all when falling off a roof. (Please don't try this at home.) We can even say that the same person's body has different weights when standing on opposite sides of the earth — same magnitude, but different direction.
So, what about the earth?
The earth has an enormous mass. Many, many, many kilograms.
The earth, regarded as a single object, is in free-fall around the sun. Therefore, regarded as a single object, the earth is weightless.
For an explanation (with a nice picture) of why an orbiting object qualifies as a falling object, see this post.
If we used a giant rocket engine to push the earth out of its orbit, then the earth would have weight while it is being pushed by the rocket, because it is deviating from free-fall. The amount of weight it would have would depend on how hard the rocket is pushing, and the direction of the weight would depend on the direction in which the rocket is pushing.
We can also regard the earth as a being composed of lots of parts — mountains, oceans, magma, and so on. If we consider just one part of the earth, say just one mountain, then that mountain has a large weight. It has a large weight because the substrate underneath the mountain is pushing up on it with just enough force to prevent the mountain from freely falling toward the center of the earth. The earth has a large mass, and mass creates gravity, so all the parts of the earth would fall toward the center if they were not prevented from doing so by pushing on each other. In this sense, each part of the earth has weight, because those individual parts are not in free-fall; the other parts underneath them (underneath = closer to the center of the earth) are pushing up on them (up = away from the center of the earth) just enough to prevent them from freely falling toward the center.
But when the earth is regarded as a single object in orbit (freely falling) around the sun, the earth is weightless overall.
The question mentioned the word "acceleration." This is worth clarifying. The word acceleration is used with two different meanings:
It is used to mean that the distance between the object and some other reference object (like the distance of a falling rock from the surface of the earth) is changing at a non-constant rate. This is sometimes called "relative" acceleration. It is undefined unless a reference is specified. When the word is used this way, the same object may be accelerating with respect to one reference and not accelerating with respect to a different reference. This has nothing to do with the object's weight.
It is used to mean that the object not in free-fall. (More technically: it is used to mean that an object is not following a geodesic in spacetime.) This does not depend on any external reference. If an object is accelerating in this sense, then it has weight.
If we carelessly use the same word with both meanings, we end up with crazy-sounding statements, kind of like this famous example: "Feathers are always light (opposite of heavy) even though sometimes they're not light (opposite of dark)."
If the earth is regarded as a single object in orbit (freely falling) around the sun, the earth is weightless. We can rightly say that it weighs zero pounds (or zero Newtons). Individual parts of the earth (like individual mountains) have weight because they are not in free-fall, and their weights are in different directions depending on where they are located on the earth. But overall, the earth is weightless, because it is in free-fall overall when regarded as a single object.
Mass is simpler. The earth's overall mass is just the sum of the masses of all of its parts. Mass does not have a direction and does not depend on how the object is moving. The earth has a very, very large mass — lots and lots of kilograms.