What is a number? A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing.  Since quantity and number mean the same thing, this definition won't do at all.
So, what is a number or quantity?
 A: What is a number? That is a good question.
If you ask the somewhat educated layman, you might get answers such as "a representation of a physical quantity". You may get some other answers too.
But since mathematicians use words from natural language with a particular meaning, let's cut the foreplay, and skip right down to the mathematician part. However there is no agreed, or even common, definition for "number". The definition I have in mind, and I suspect that many mathematicians would agree with me, is the following one:

We say that $x$ is a number, if it is an element of a number system, which is a system representing and measuring a quantity of some form.

This definition allows for natural numbers, integers, rational numbers, real numbers, complex numbers, ordinals, cardinals, and so on. All these are number system. The only thing they have in common is that they measure some sort of quantity, and they represent it somehow.
Therefore, for me, the context "X numbers" means that "X" is some sort of form of measurement for mathematical objects.
A: The word "number" means what I wish it to mean, nothing more, nothing less.
Depending upon what I'm doing, things I've wished "number" to mean have included:


*

*Natural number

*Integer

*Rational number

*Real number

*Projective real number

*Extended real number

*Complex number

*Projective complex number

*Hyperreal number (and the nonstandard versions of all of the above)

*Continuous real-valued function

*Ordinal number

*Cardinal number

*Polynomial over a finite field

*Rational function over a finite field

*Element of a particular ring I'm working with

*Abelian group

A: In set theory, we begin by defining $0$ as the number of elements in the set having zero unique elements.  Then, for each subsequent number that we wish to define, we say that $i+1$ is the number of elements in the union of the $i$th set and the set containing the $i$th set.  If we label this series of sets as $u_i$, then we have $u_0=\varnothing$, $u_{i+1}=u_i\cup \{u_i\}$.  Then our first few "numbers" are as follows:
$$u_0=\varnothing$$
$$u_1=\{\varnothing\}$$
$$u_2=\{\varnothing,\{\varnothing\}\}$$
$$u_3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$$
...
This is rather like getting something from nothing, but it definitely gives uniqueness to each integer.
A: This is a never interesting question and is remarkably complicated. The reason for the complexity is two-fold, there are many different number systems, and constructing them is a lot of work. For example, the most popular way to define natural numbers  defines $2=\{\{ \}, \{\{ \} \} \}$. The number $\frac{1}{2}$ is a set containing $\{(1, 2),(2, 4), (3, 6) ...\}$ (It's all the numbers $p/q$ such that $2p=q$, but then you have to define multiplication (This is done with the Peano Axioms)). Real numbers are even more complicated, and can be defined as all the rational number before the real number itself (A dedekind cut). In the end, if you want a satisfying answer to this question, you need to learn a lot of math. Then again, there are multiple definitions of each, and few mathematicians really ever use them unless there doing a proof involving the construction itself. 
If your interested in learning, Naive Set Theory by Paul Halmos and Real Analysis by Pugh are a good start.
A: The concept of "number" according to mathematicians has expanded dramatically over the centuries. It's by now very hard to give an all-encompassing definition. 
There is a book Numbers written by a group of German mathematicians, with some historical notes, and broadly speaking in three parts. Under their Part A, we have Chapter 1: Natural numbers, integers, rational numbers. Chapter 2: Real numbers. Chapter 3: Complex Numbers. (...) Discussion of algebraic numbers. (...) Chapter 6: The $p$-adic numbers (and I suppose we ought to include local fields in the sense of Weil, and also adeles). 
Under Part B, we head off into real division algebras; we have Chapter 7 on Hamiltonian numbers or quaternions. Chapter 9: Cayley's numbers or octonions. One could go on to composition algebras (chapter 10). 
Under Part C, we head off into more set-theoretic territory. There are the models of nonstandard real numbers inaugurated by Abraham Robinson (chapter 12). Then there are Conway numbers or surreal numbers (chapter 13). Chapter 14 discusses cardinal numbers and ordinal numbers. 
In most of these systems, there are operations of addition and multipication so that most of these number systems can at least be described as rings or algebras of one sort or another, in the senses that mathematicians give these terms. 
A: To my view and knowledge with referring some Frege's arguments: A number is a least common property that objects/things of the same "quantity"(in a primitive sense) have to share.
A: It depends on the context. The word number could for example mean integer, rational number, real number or complex number, all of which have precise definitions. In some situations it could even mean something like "an element in some particular ring or field", again this is well-defined. There are lots of other "number systems" in use as well, and it's probably impossible to list all of them.
The problem with a dictionary definition is that they don't build the language using undefined terms, axioms and definitions.
A: I love this question.
Most people's definitions of number seem to involve the idea of quantity. I guess this derives from what we presume to be the origins of numbers  - the counting of things like goats. But I think this is misleading and ultimately a dead end...
Quantity seems to be left behind quite early on in the development of numbers. As soon as we start using irrational numbers we have things regarded as numbers which do not relate directly to quantity - you can't have Pi of something or the square root of 2 of something.  When we get to complex numbers the connection to quantity seems more remote still.  If we had started with complex numbers instead of natural numbers the idea that numbers related to quantity would seem far fetched indeed.
Having given it some thought the only consistent theme I can find between all the generally accepted uses of the word number is that of 'constancy': a number is the name we give to a constant; that is why Pi is a number: Pi is the name we give to the constant ratio between the area of a circle and the square of its radius.  Any value of any numbering system expresses some quality of unchangingness.
So my definition of a number system would be a 'symbolic system of constants' and a number would be a member of such a system.  Natural numbers are thus the set of symbols which represent all possible states of discreet existence(or something like that).
A: This is rather an extended comment than answer, but like many (good) existing answers, I do want to challenge the idea underlying the question, and expressed in your comment from Sep. 16, 2013,, that a concept like "number" is necessarily given by some common definition, i.e. all the things we call "number" have to have some clear thing in common.
Actually, this very notion has been famously discussed by Wittgenstein in his Philosophical Investigations, especially paragraphs 65--77, where instead he proposes the idea that some concepts are given by "family resemblance":
https://en.wikipedia.org/wiki/Family_resemblance
(This WP article seems to be a good first introduction, and it discusses similar ideas other philosophers have or have had; but I really recommend reading at least the above paragraphs in Wittgenstein's original text. Then again, I recommend reading the entire Philosophical Investgations whenever I can.)
Actually, it turns out that "numbers" are the next example (after his famous introductory example "games") that Wittgenstein explicitly talks about:

for instance the kinds of number form a family in the same way. Why do we call something a "number"? Well, perhaps because it has a direct relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.

-- loc. cit., §67
