Avogadro's Number is a very large number approximately equal to this much:
It's a useful number because it helps scientists figure out how many atoms or molecules are in a sample of substance, just by weighing the sample. (Provided you also know what kind of atoms the substance is made of.)
The idea for this number started with experiments done by chemists. Chemists knew, for example, that Carbon Monoxide consisted of 1 carbon atom attached to 1 oxygen atom. And they also knew that the weights (masses) of the carbon and oxygen samples were in the ratio of 3-to-4. In other words, 3 grams of carbon always mixed with 4 grams of oxygen; or 6 grams of carbon with 8 grams of oxygen; or 30 grams with 40 grams, and so on. In every case, to make a sample of carbon monoxide, the required amount of oxygen always weighed a little more than the amount of carbon (4/3 as much, in fact).
So from experiments like that, chemists concluded that an oxygen atom must weigh 4/3 as much as a carbon atom. By doing lots of similar experiments, they were able to assign relative weights to every kind of atom. So they eventually knew, for example, that hydrogen was the lightest of all atoms; that a helium atom was 4 times as heavy as a hydrogen atom; a carbon atom was 12 times as heavy; an oxygen atom was 16 times as heavy, and so forth. But they only knew the RELATIVE weight between one kind of atom and another. They didn't know how many actual "grams" were in one atom of hydrogen. To put it another way, they didn't know how many atoms were in a 1-gram bottle of hydrogen.
But they did know this: since they knew that helium atoms were 4 times as heavy as hydrogen atoms, they knew that if you took a 1-gram bottle of hydrogen, and set it next to a 4-gram bottle of helium, the two samples must have the SAME number of atoms. Likewise, you could line those up against 12 grams of carbon and 16 grams of oxygen; then you would know that each of those four bottles held the SAME number of atoms.
They didn't WHAT that number was. But they made up a name for it. They called it "a mole." To get "a mole" of some substance, you look up its "atomic weight" in a chart (basically, the relative weight of 1 particle of that substance), then you measure out that many grams. A mole of one substance WEIGHS a different amount than a mole of another substance, but they both have the same NUMBER of particles (atoms or molecules). This was useful information, because then they could describe chemical reactions in terms of "moles." They could say, "mix 2 moles of hydrogen atoms with 1 mole of oxygen atoms, and you get 1 mole of water molecules."
But they still didn't know how many molecules that meant. Eventually they were able to do experiments (I won't describe them) that determined that 1 mole (of anything) equals 602,214,150,000,000,000,000,000 particles. (Interestingly, Amadeo Avogadro didn't do those experiments himself; but other scientists decided to name the number after him, in his honor.)