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 $\infty$?

Just to be clear: Infinity is not a number.

Also, it is likely that there is no “exact” (agreed upon) characterization of infinity.
In a sense, casually put: $$∞={that which is NOT finite}\infty = \{\text{that which is NOT finite}\}$$

Definition: Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latin infinitas, which can be translated as “unboundedness“, itself derived from the Greek word apeiros, meaning “endless“.

“In mathematics, “infinity” is often treated as if it were a number (i.e., it counts or measures things: “an infinite number of terms”) but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.”

For a more expansive discussion on infinity, y

EDIT: You might want to be assured that you are not the only one grappling with the concept of infinity: Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism.