What are Numbers? Hi Qwex, I really liked that you question the concept of numbers.
I believe that when you try to search for the origin of words and the way in which we pour meaning into the world around us - it is most fruitfull to search how those meaning are first discovered, whether it is understanding how a baby generates those concepts or reading the history of how those concepts were created in the early human culture.
A child starts by counting objects in the world - the concept of numbers as something other than an adjective to a group is outside the scope of our first usage of them. This is on par with how we evolve language itself - we use sounds and words that allow us to interact better with the world around us.
In the same way the usage of numbers in the human history evolved as a way to describe actual acts in the world - ways to divide a field of wheat, to make commercial transactions, etc...
It was then used as a way to allow us to make astronomical inquiries - taking part in one of the most early abstract human endeavours: religion. The Pythagorean sect in ancient Greece even evolved mathematics as a religion of its own. It is not weird than that Geometry, so closely related the world of mathematical usage in those days, was the tool by which the Pythagorean advanced their understanding of the world. And an ancient greek would asked then, in the same way you ask now - why do we talk about lines, what are lines?
So my answer to what are numbers - they are signs (verbal/graphic), like words, by which we are able to interact better with the world. The scope in which we use them is sometimes confined to the abstract term - mathematics; But this term, describing a narrower usage of them, isn't all that we do with those signs. We use numbers to describe concepts in every field of human knowledge.
And for a more concrete terminology - they are most widely used to describe a quality of a group of objects, distinguishing that group by the count of object within it. You can basically grow all of the mathematical knowledge from that definition (given a set of logical axioms on such groups).