Are some real numbers "uncomputable"?

As you have observed yourself, $\mathbb{R}$, the set of real numbers, is uncountable, but every recursively enumerable set is necessarily countable. This immediately implies that there exist uncomputable real numbers. The same argument shows that there are (formally) indescribable real numbers. Indeed, almost all real numbers are indescribable, and a fortiori, uncomputable. There is nothing wrong about this, though it may be disturbing.

Do uncomputable/indescribable real numbers ‘exist’? Well, that's a philosophical question. A Platonist might say that they exist, even though we have no means of naming them specifically. A finitist might say they don't exist, precisely because we have no algorithm to compute or even recognise such a number.

Does this impact the way we do mathematics? Not really. So what if the vast majority of real numbers are uncomputable? By and large we deal with generic real numbers, not specific ones. For example, the fact that every non-zero real number $x$ has an inverse does not rely on the computability properties of $x$.

I think you are talking of computable numbers. You can read more about it on Wikipedia. Unfortunately, what I know regarding this is only a very little subset of what Wikipedia has. A relatively famous example for a non-computable number is the Chaitin's constant.

Most real numbers are not computable. Informally speaking, a real number is computable if there is a machine or algorithm that computes its decimal expansion, one digit at a time, that is, you can ask for the $n$-th digit. Once you formalize machines and algorithms, which are finite animals, you see that there are only a countable number of them and so there are only a countable number of computable real numbers. Since the real numbers are not countable, most real numbers are not computable.