one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....).
there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals.
as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity.
even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming.
* the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.
A better way to count based on names would be to first count all single letter names alphabetically, then all two letter names alphabetically, etc.
Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"
the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings.
Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number.
for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place.
`+` :
technical details include that some numbers have non-unique representations as infinite sequences of digits.
For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going).
if this irritates you, let z = 0.999...
then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1