What does research in theoretical computer science involve?
Regarding topics

There are different type of issues that may be considered
theoretical computer science. The important word here is
"theoretical" (as we all have some idea of what computer science
deals with). Understanding the word theoretical is not so obvious.
For a long time I took it to mean mathematical, as opposed for example
to "hacking". I learned better from people in linguistics:
theoretical for them clearly does not meant mathematical, but based
on a theory which may be somewhat informal (though it may be also
mathematical), and is an organized body of knowledge and concepts that
structure understanding of observable phenomena and hopefully allow
some deductive and predictive use of the acquired knowledge. It also
reduces the amount to learn and to teach by reducing the number of
primitive concepts from which the rest can be deduced.

Theoretical can be opposed to practical, which is how this knowledge
is used to actually run computing engines, to build systems, etc. I
can also be opposed to applied which is the use of this knowledge to
address problems in other fields of science and human activities.

But I doubt there are clear-cut boundaries.

This said, theoretical computers science covers diverse domains, and I
will try to give some, while I am sure I forget others, and also that
other people may disagree with this organization.


one domain is computability, which studies what can be computed,
and how in a rather abstract sense: largely what is described in
Suresh Venkat's answer.
another is algorithmics, that finds effective means to compute
answers to specific problems, with specific
constraints. Computability is a theoretical context for algorithmics.
semantics (for want of a better name), analyzes the conceptual
organization of computational problems, and of algorithms, into
higher level concepts, so as to factorize techniques that have
proved useful and are often reused, such as the concept of
subprogram,data-structures, modules, information hiding. It
includes the development of mathematical tools that formalize
adequately these concepts to allow high-level reasonning (Scott
semantics for example). It also touches on the way this is
expressed, thus on the separation and relation between syntax and
semantics. Programming languages concepts are part of it (though
language design is probably the pratical application of that
knowledge). It can also include the relation between proof theory
and computation theory, and the modern role of type systems.
another topic, which could develop more than it has so far, is the
relation between computation and fundamental physics. For
example. is there a relation between the limits on computation and
the properties of the physical world, such as physical information
density or the laws of thermodynamics. Quantum computing may
improve a bit our computational prowess; could we hope for more?
Some may dispute that this is still TCS, though there are TCS
studies on hypercomputation.


Regarding specific activities

I am skipping the obvious activities required by academic life. or scientific life in industry:
teaching, publishing, reviewing papers, writing grant applications,
taking academic responsibilities, managing people, advising students
or policy makers. But even then, there is no simple answer to your
question. Here I am just sketching a few aspects that come to mind,
but I am sure there is a lot more to be said. And I am not sure I am
answering you. Some of the best scientists have written books about their work, and that may give you hints about scientific activities.

Researching in theory may involve a variety of things, depending on
your skills and interests that vary a lot from scientist to
scientist. It is somewhat hard to talk of it, since each person
perceives more her own activity and interests than that of others. Most reasearch
requires a wide knowledge, since interesting and really original
results often comes from putting things in relation, or transferring a
technique from one (sub)field to another, or getting different
technical views of the same problem. So learning as much as you
can in breadth as much as in depth is important. Remember that while
you have the time and ability for it as a student, or as a junior
faculty/scientist, both will be reduced later, because of responsibilities
and life in general. Teaching what you do not specialize in may be a
way to keep learning. On SE you can probably learn more by answering than
by asking.

The kind of work people do can vary a lot, because people are
fortunately very different, with a great variety of interests and
technical abilities, thus complementing each other.  Some people are problem solvers. They look at
theoretical or practical questions raised by other people, or by
themselves, and try to solve them, or get closer to a complete or
partial solution.  Other people will be better at structuring existing
knowledge, and putting thing in relation, and then finding new
questions to ask. Both are essential.

Finding simpler proofs of technical results, or simpler presentations
of theories, or merging concepts is important. It generalizes results,
reduces the numbers of things to learn, emphasizes the essential
ideas and possibly brings new understanding. Since our learning time
is limited, progress is possible only when we condense knowledge.

A simple example is the study of abstract families of formal
languages. When language theory first developped, closure properties
under various operation were proved again and again for each family of
languages (regular, context-free, RE, ...), with ad hoc techniques
depending on the family. Then it occurred that these closure
properties had intrinsic relations independently of the concerned
families, and they were studied as such. Today, we only have to check
a few of the simpler closure properties for a given family, and we
get "for free" a whole set of other properties.

Another important point is that there is not such a clear-cut distinction
between theoretical, practical, or experimental work. A good theory
may lead to the implementation of systems that can mechanize the
resolution of problems. And it will take a good theoretician to
implement such a system, with a mix of theoretical and practical work,
including system implementation, or language design. Many examples
come to mind, such as proof and/or program synthesis systems,
specialized language for synchronous parallel systems, a restricted
algorithmic language for which computational complexity can be
systematically derived. Not only is it important to be able to produce
such pratical systems, which make theoretical results more widely
available and usable, but it is often very important for theoretician
either to use proficiently these systems, if only to unload the now less creative parts of his work, or to contribute to the development and
extension of these system.

Another aspect is to be able to compare theoretical approaches by
pratical experimentation. Here, the issue is to compare different
techniques to accomplish the same goal. Comparing
implementations is often meaningless as their efficiency often depend
on the programming language, or the programming skills of the
implementor. But if they can be expressed all in a common theoretical
framework, then it is sometimes possible to compare them experimentaly
within that framework. Here, theory and practice support each other,
as they often do in science. Pure theoretical analysis is not always
easy to achieve. Furthermore, experimental analysis, when well
conducted, can exhibit unexpected behavior that may motivate better
theoretical analysis.

The world is not simple or clear-cut. That is why it can be fun, with
room for all kinds of skills. Questionning your own knowledge, and
answering questions of others, by whatever means.

Two things often forgotten: ethics of science, and explaining it to
people.  Both are essential, and hard.