The answers so far seem pretty good, but I'd like to try a slightly different angle.
Before I get to atomic orbitals, what does it mean for an electron to "be" somewhere? Suppose I look at an electron, and see where it is (suppose I have a very sophisticated/sensitive/precise microscope). This sounds straightforward, but what did I do when I 'looked' at the electron? I must have observed some photon that had just interacted with that electron. If I want to get an idea of the motion of the electron (no just its instantaneous momentum, but its position as a function of time), I need to observe it for a period of time. This is a problem, though, because I can only observe the electron every time it interacts with a photon that I can observe. It's actually impossible for me to observe the electron continuously, I can only get snapshots of its position.
So what does the electron do between observations? I don't think anyone can answer that question. All we can say is that at one time the electron was observed at point A, and at a later time it was observed at point B. It got from A to B... somehow. This leads to a different way of thinking about where an electron (or other particle) is.
If I know some of the properties of the electron, I can predict that I'm more likely to observe an electron in some locations than in others. Atomic orbitals are a great example of this. An orbital is described by 4 quantum numbers, which I'll call $n$, $l$, $m$, $s$ (there are several notations; I think this one is reasonably common). $n$ is a description of how much energy the electron has, $l$ describes its total angular momentum, $m$ carries some information about the orientation of its angular momentum and $s$ characterizes its spin (spin is a whole topic on its own, for now let's just say that it's a property that the electron has). If I know these 4 properties of an electron that is bound to an atom, then I can predict where I am most likely to observe the electron. For some combinations of $(n,l,m,s)$ the distribution is simple (e.g. spherically symmetric), but often it can be quite complicated (with lobes or rings where I'm more likely to find the electron). There's always a chance I could observe the electron ANYWHERE, but it's MUCH MORE LIKELY that I'll find it in some particular region. This is usually called the probability distribution for the position of the electron. Illustrations like these are misleading because they draw a hard edge on the probability distribution; what's actually shown is the region where the electron will be found some high percentage of the time.
So the answer to how an electron "jumps" between orbitals is actually the same as how it moves around within a single orbital; it just "does". The difference is that to change orbitals, some property of the electron (one of the ones described by $(n,l,m,s)$) has to change. This is always accompanied by emission or absorption of a photon (even a spin flip involves a (very low energy) photon).
Another way of thinking about this is that the electron doesn't have a precise position but instead occupies all space, and observations of the electron position are just manifestations of the more fundamental "wave function" whose properties dictate, amongst other things, the probability distribution for observations of position.
Based on some of your comments, I think what might be tripping you up is the first statement you started with:
From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels.
and
...If suppose, we assume electron losses total energy, electron can't stay in any particular shell, as it would not have that particular value of energy.
That may be true for Bohr's atomic model, but Bohr's atomic model is wrong. And electron does not have to be in a particular, definite shell or energy level. Rather, any electron state is a superposition of states of definite energy level (energy eigenstates).
That means the expectation value of a hydrogen electron state is going to look like
$$\langle E\rangle = \sum_n |a_n|^2 E_n\text{,}$$
where $\{a_n\}$ are arbitrary complex values with $\sum_{n>0}|a_n|^2 = 1$ and $E_n$ are energy levels in increasing order. Because of the sum-to-$1$ condition, taking any portion along the other energy eigenstates will increase energy compared to the ground state.
In other words, even if the electron state does not have a definite energy, you still can't go lower than the ground state.
Suppose, I have a cup of hot coffee on the table. It will be continuously losing energy in the form of heat, but it stays on the table, though there was a energy loss. Now, all of a sudden, I take off the table, the cup of coffee converts it potential energy into kinetic energy to come down.
If you don't shake the table, the coffee cup will sit there, forever. Similarly, nothing perturbs the electron in an excited energy eigenstate, then it simply will never decay. It cannot: energy eigenstates are stationary; they do not evolve into anything other than themselves.
However, being completely without external perturbation is actually impossible. The uncertainty principle provides the electromagnetic field with vacuum fluctuations, which will perturb the electron even if nothing else in the environment does. In your analogy, this (or something else) provides the "shaking of the table" for the electron. Once the electron state gains even a tiny component in some other energy eigenstate, the state can evolve in time.
In other words, one can think of spontaneous emission as a particular type of stimulated emission where it's the vacuum that does the perturbing.
Best Answer
Electrons are quantum mechanical entities,they cannot be described by the motions of classical billiard balls. That is why the terminology "orbital" was invented. We only know about the electron around the proton in the hydrogen atom that when probed, it has a probability of being within the orbital solution.
As far as the mathematics goes there is only the probability and no continuity between two space points so that a classical velocity could be defined.
The electron remains at its energy level, unless a photon with the appropriate energy hits the atom. Orbitals might mathematically overlap in space, particularly if there are many electrons, this does not mean there is an energy exchange, it just means that the probabilities of two energy orbital can overlap in space. Only if there is a probability for an energy transfer to a lower state then the space overlap has a meaning. This is how electron capture happens with nuclei , the S level overlaps with the nucleus and there exists a probability for the electron to be captured by a proton given the energy balances of the system.
See above. The electron is not moving like a billiard ball. Its path is not consecutive. There are only probabilities.
There is nothing instantaneous in quantum mechanics . If one wants to go to the details of "electron interacting with a photon" one has to go to quantum field theory, where the interactions are described with Feynman diagrams and there are constants characterizing them which define the order of magnitude of the time transition.
No , there are no infinities . If the electron is in an energy level that can release a photon and relax to the lower level, there exist time constants coming from the probability distributions which give a lifetime for the decay and a width to the energy line.
If it is at the ground state it will stay in the ground state forever, unless an appropriate energy photon hits the atom, either to kick it up an energy level or free it from the material completely, as in the photoelectric effect.