Gravitational force and Electromagnetic force? I found this interesting note in one of my textbooks,

The enormous strength of the electromagnetic force compared to gravity is evident in our daily life. When we hold a book in our hand, we are balancing the gravitational force on the book due to the huge mass of the earth by the 'normal force' provided by our hand. The latter is nothing but the net electromagnetic force between the charged constituents of our hand and the book, at the surface in contact. If electromagnetic force were not intrinsically so much stronger than gravity, the hand of the strongest man would crumble under the weight of a feather! Indeed, to be consistent, in that circumstance, we ourselves would crumble under our own weight!

What does this mean?
 A: The notion of "weak gravity" is "not even wrong"! It's comparing oranges to apples: it's meaningless to compare a dimension-full interaction (gravity) with a dimensionless interaction (standard model interactions such as the electromagnetic force), if no circumstance (e.g the specific charges and masses in comparison) is provided.
The Nobel laureate Frank Wilczek actually wrote a whole book ( The Lightness of Being) about what should be the right question: Given that the "charge" of the gravitational force is mass (energy tensor), the correct question is why are the masses of the elementary particles so small compared with the Planck scale?
This leads you to the nagging issue of natureness/hierarchy problem (the unnatural gap between the Planck mass and the weak scale/Higgs mass).
As of yet, mortal physicists are still scratching their heads and fretting about this nasty "naturalness/hierarchy/fine-tuning problem". The world’s best minds are loosing sleeping on it (ask Lisa or Nima) and yet there is no answer.

To give you some thought experiment of "strong" gravity:

*

*If the electron mass is increased to the mass of a flea egg ($10^{-8}$ kg,
the plank mass), the gravitational attraction between electrons will
be in balance with the repulsive electronic force. In technical
jargon, the Schwarzschild radius and the Compton wavelength are of
the same order for this case.


*If the electron mass is increased to the mass of a chicken egg, the
gravitational force between electrons will trump the electronic force
and electrons will be crushed into an electron black hole by gravity. Of course,
true quantum gravity effects will be the dominant one in this case, the conventional semi-quantum black hole reasoning (a la Stephen Hawking) shall be taken with a grain of salt.
A: I think the part about electromagnetic force being exponentially stronger than gravitation got through to you!(An easy way to picture this is how static electricity can lift up things against the pull of the earth!) About the feather,the way you are able to support the feather against the gravitational pull of the earth is because of the electromagnetic forces exerted by your hand on the feather and vice versa. If these were much much weaker something as little as a feather would be able to crumple your hand, that is if it hadn't already been crumpled by gravity before that!
I've read in The Briefer History of Time (Stephen Hawking), that even very slight changes in the strength of the fundamental forces can cause great imbalance. If the electromagnetic force was stronger, atoms wouldn't be as stable as they are, and if it was weaker, atoms might not form at all, so the chances of life existing would be zero! 
If there was a sudden change, hypothetically, this instant, the whole world could possibly crumble. But if that actually was the case we wouldn't even have been here.
A: The weak, electromagnetic, and strong fundamental forces can all be quite important, depending on the energy and distance scales of the phenomenon in question. How about gravitation? It is negligible, at least for what concerns the energies we can reach in particle accelerators (in fact, gravity is not even included in the  the Standard Model). Gravity manifests its power only at the macroscopic scale or at the unreachable Planck  scale.
This "puzzle" of "weak gravity" is an unsolved problem in theoretical physics, known as the Hierarchy Problem.
You can easily see that gravity is really "weak" at the fundamental level: in the static limit, the electric and gravitational force laws are both inverse square laws, so if one computes the ratio of the forces between two bodies, the distances cancel.
Take two point particles with masses $m_1$, $m_2$ and charges $q_1$, $q_2$. You have the gravitational  force and the electric Coulomb force (CGS units are used)
$$
F_G = G \frac{m_1 m_2}{d^2}
\qquad \quad 
F_C = \frac{q_1 q_2}{d^2}
$$
Clearly (depending on the actual values of $m_1$, $m_2$, $q_1$ ,$q_2$) you may have that $F_G > F_C$, or the other way round.
However, for (charged) elementary particles, you always have that $F_G \ll F_C$. In particular, if you use two electrons  ($m_1=m_2=m_e$,
$q_1=q_2=q_e$), then you have
$$
F_G / F_C =  \frac{G m_e m_e}{q_e q_e} = G \left(\frac{m_e}{q_e} \right)^2 \sim 10^{-42}
$$
where the quantity in the parentheses is the mass to charge ratio first measured by Thompson. As you see $F_G \sim 10^{-42}F_C$, independently on the distance $d$: the gravity attraction is really extremely weak.
In an Hydrogen atom you should compare the attraction between the electron and the proton (i.e. the Hydrogen nucleus). The proton is $\sim 2000$ times more massive than the electron so $F_G$ is $2000$ times bigger than the electron-electron gravitational attraction considered before. In this case you should find $F_G/F_C \sim 10^{-39}$. Still a super small number, telling us that there is no need to include gravity when studying the atomic structure.
Clearly if you don't use fundamental particles but, say, planets (that are almost neutral but have a huge mass), then $F_G$ wins. However, using planets and stars is, clearly, not in the spirit of the Hierarchy problem.
A: This sort of thing just compares the strength of gravity to the strength of EM. It is a bit of a "wow" factor in popular science to say that all the earth's gravity is resisted by a few electrons as you hold something above the ground or whatever. As the top commenter pointed out, this is technically comparing apples to oranges, and the right question is not "why is gravity so weak" but rather "why are things so light" instead. That may be above your needs though.
