I have seen all the seasons of The Big Bang Theory but one thing that's bothering me was what Leonard really does at the California Institute of Technology. Is he a teacher or doing a research which is not gonna get over?
Leonard is an experimental physicist, which means that he designs and conducts experiments (often with lasers) to prove or disprove theories put forth by other physicists, like his roommate Dr. Sheldon Cooper (a theoretical physicist).
We see Leonard carrying out a range of research as an experimental physicist. Initially his work seems to have revolved around confirmatory research on the work of other scientists (with a strong emphasis on work involving high energy lasers) but in more recent shows, he's been shown to be engaged in increasing amounts of primary research, especially in the area of black holes, particle research and dark matter, publishing regularly and evidently catching the eye of other physicists such as Stephen Hawking.
In the most recent two seasons, he and Sheldon (a theoretical physicist) are being lauded for their work on combining superstring and dark matter theory.
...right now I’m designing an experiment to study the soft component of cosmic radiation at sea-level
... I’m attempting to replicate the dark matter signal found in sodium iodide crystals by the Italians.
... I was up all night using the new free-electron laser for my X-ray diffraction experiment.
my tests of the Aharonov-Bohm quantum interference effect have reached an interesting point. Right now, we’re testing the phase shift due to an electric potential.
... an expedition to the North Sea to test hydrodynamic simulations of black holes. One of their experimental physicists dropped out, and I recommended you.
Alex: So, what kind of research are you doing?
Leonard: High-energy lasers.
... Can you believe there was a time when we would have needed an array of giant Cherenkov telescopes to detect cosmic particles? And here we are building our own multi-wire detector like a couple of badasses.
It is likely that Leonard, Sheldon and Raj are all on post-doctoral research posts, extended because they have applied for more funding from external bodies. They have spoken about funding during the show, indicating that their projects are funded by research grants. These are always fixed-term, so it's likely that the characters would all have to apply for new funding every few years in order to keep their jobs.