What exactly is infinity? On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and $\infty - \infty$.
From what I know, given $x$ being any number, excluding $0$, $\frac{x}{x} = 1$ is true.
So just what, exactly, is $\infty$?

Cantor demonstrated that there could be varieties of infinity. Different varieties of infinity have different properties - ordinals and cardinals for example both have infinities (which can be related to each other), but have different arithmetic.
Then there are infinities which come from constructions like the Riemann Sphere (which compactifies the Complex numbers by adding a single point at infinity). This is in contrast to projective geometry in which lines and planes at infinity can be introduced to affine space (eg "ordinary three-dimensional space").
Infinities are useful in various ways in mathematics - hence the variety of constructions illustrated above - and can even be given arithmetic properties, analytic properties and geometric meaning. However, the particular meaning and properties of infinity will depend on the mathematical context.
And note that using infinity brings losses as well as gains - we cannot retain all the properties we had before infinity was introduced.
Infinity is not a natural number, or a real number: there should be no confusion about that. We can use infinity as the upper limit of an integral as shorthand to say that all the reals greater than the lower limit are included - that is a conventional use - along with others involving arbitrarily large numbers. Really this is a shorthand for some kind of limit process, but a shorthand which is convenient and useful.
We can augment the Reals or the Naturals in potentially useful ways to include a number called $\infty$ or $\omega$. But then the augmented Reals, for example, cease to be a complete ordered field and adding a single infinity to the Naturals potentially brings with it a host of others.
Mathematicians have ceased to be scared of infinity over the years, and have used ideas of infinity in a range of context to express important mathematical ideas.
