What do researchers do on a daily basis? They teach classes (at all sorts of levels, in principle, and depending a bit on what kind of institution they are at
or what country they are in, from beginning undergraduate to advanced graduate level), they advise thesis students (some mixture of undergraduate thesis students, masters thesis students, and Ph.D. students), attend research seminars (where speakers report on their latest research), and do their own research.
"Doing their own research" ranges from [here I am speaking about pure mathematical research, which is where my own experience lies]: studying other books and papers to get inspiration, learn new techniques, or look for a specific tool which could help them with a problem they are confronting; working out their own new ideas, perhaps just sitting by themselves with pen and paper, or perhaps in discussions with a collaborator; writing up papers explaining their new theorems; or many other possibilities (e.g. writing computer code to investigate certain situations or phenomena).
As for what the new results are that people prove, here one can say that in most fields of mathematics, there is a broadly understood and acknowledged (to the workers in that field) "frontier" of cutting edge knowledge, and some sense
of what things that are not currently understood are important and should be investigated. People then try to push their way over that frontier into these unknown areas in some promising direction, with the hope that they will push the frontier a little way further. (Often there are very famous open problems in a field, such as the Clay Millenium Prize problems, but most people are not working directly on such difficult problems --- those problems are difficult and celebrated for a reason. Rather, people work typically work on more tractable,
and incremental, problems, with the hope --- which seems borne out by the overall experience of many generations of mathematicians --- that consistent incremental progress in pushing the frontiers of understanding forward adds up over time to significant advancement of our knowledge.)
It is hard to be more specific about the kinds of theorems that people prove,
because they are incredibly varied in subject area (the umbrella of "pure mathematics" is a very large one), and it is also often hard to explain cutting edge
mathematical research to someone who doesn't have some background knowledge in the subject area. There are some famous results that you have probably heard of (e.g. Wiles's proof of Fermat's Last Theorem or Perelman's proof of the Poincare conjecture). As I noted above, most people's research results are typically on a smaller scale than these, but it's not unreasonable to take them as models, and just to imagine that people are proving somewhat similar things (albeit a little more specialized and less celebrated): perhaps proving properties of certain classes of Diophantine equations, or of certain kinds of manifolds.
If you go to the Arxiv, and click on the link for new mathematics papers, you will see lots of new mathematical research papers, on
a huge array of subjects. The terminology will probably all look incomprehensible, but perhaps it won't look as boring as your current mathematics syllabus. (Certainly, people who go on to pursue a research career don't find mathematics boring; it is normally the opposite --- they do it because they find it too interesting to do anything else.)