What does it mean to say "Gravity is the weakest of the forces"? I can understand that on small scales (within an atom/molecule), the other forces are much stronger, but on larger scales, it seems that gravity is a far stronger force; e.g. planets are held to the sun by gravity. So what does it mean to say that "gravity is the weakest of the forces" when in some cases, it seems far stronger?
 A: I don't think that any of the existing answers fully answer this rather subtle question (correctly). If we just consider the interactions themselves, and not particular particles that they couple together, then there is no meaningful sense in which gravity is any weaker than any of the other forces. This simply follows from the fact that the gravitational coupling constant $G$ has different units than the coupling constants of all of the other Standard Model interactions, so the fundamental "strengths" of the interactions are incomparable.
It's true that for some applications, it's simplest to work in Planck units where $\hbar = c = G = 1$ - but that statement simply reflects the fact that no dimensionless ratios can be formed out of those constants, so this simultaneous assignment is possible. It's therefore incorrect to say (as claimed in another answer) that the strength of gravity is "equal" to the strength of any other interaction in any nontrivial way. It's simply incomparable, neither weaker nor stronger. You could just as easily choose a system of units (like SI!) in which either coupling constant's numerical value is arbitrarily larger than the other's.
To meaningfully compare the strength of gravity to that of the other interactions, you need to consider the specifics of the Standard Model (SM) matter fields. The dimensionally meaningful statement is that $G M_H^2/(\hbar c) = (M_H/M_P)^2 \approx 10^{-34} \ll 1$, where $M_H$ is the Higgs mass and $M_P$ the Planck mass. (You get similarly small numbers if you plug in the mass of any other SM particle.) Different types of physicists will find different natural ways to interpret this inequality.
To a (phenomenological) particle physicist, the natural mass scale is the Higgs/SM scale $M_H$ (and the natural velocity and action scales are $c$ and $\hbar$). From this perspective, $G = M_P^{-2} \approx 10^{-34}$, and the natural question is

"Why is the gravitational interaction between SM particles so much weaker than the other SM interactions? Or equivalently, why is the Planck mass so huge relative to the SM scale?"

To a quantum gravity theorist, who is less concerned with the details of the Standard Model, the natural mass scale is the Planck mass $M_P$. From this perspective, $M_H \approx 10^{-17}$, and the natural question is

"Why is the Higgs mass so tiny relative to the Planck scale?"

This is what Wilczek meant when he said "The question is not 'why is gravity so weak?' The question is 'why is the electron mass so small?'." I'll leave it to the philosophers to debate whether this is actually a "better" formulation of the question. The "hierarchy problem" really encompasses both formulations of this question, but is usually formulated from the latter perspective, and phrased in terms of "unnaturally fine-tuned radiative corrections to the Higgs propagator" and a bunch of other jargon involving the renormalization group, whose details are orthogonal to the OP's question.
Another answer states another common misconception, which is that the weakness of gravity simply stems from the irrelevancy/nonrenormalizability of the gravitational interaction. To see why this explanation is seriously incomplete, it's useful to consider yet another type of physicist's perspective: that of a condensed-matter theorist. The key point is that irrelevant operators (in the technical sense of the word) are only irrelevant (in the colloquial sense of the word) at energies far below the microscopic "UV cutoff" energy scale.
In this case, the cutoff scale is the Planck scale, so it is indeed straightforward to show from general arguments that gravity is very weak at energies far below the Planck scale. But this doesn't really answer the question; it just pushes it back to the question "Why do SM particle interactions occur at energies so far below the 'natural' Planck scale?" In a generic system, we would expect the low-lying excitations (the elementary particles) to have mass gaps on the order of the microscopic energy scale (the Planck scale). Only very close to a phase transition does the mass gap almost close and field theory become applicable. So to a condensed-matter physicist, the natural formulation of the hierarchy problem is

"Why are the Planck-scale interactions so finely tuned to lie near a phase transition, thereby allowing us to use field theory to accurately describe the low-lying excitations (the elementary particles)?"

A: Gravity seems stronger because it's always attractive. Of the other 3 interactions:


*

*Electromagnetism has positive and negative charges, so it only manifests macroscopically when there is a charge imbalance.

*The weak and strong interactions are intrinsically short-ranged.

A: When we ask "how strong is this force?" what we mean in this context is "How much stuff do I need to get a significant amount of force?" Richard Feynman summarized this the best in comparing the strength of gravity - which is generated by the entire mass of the Earth - versus a relatively tiny amount of electric charge:

And all matter is a mixture of
  positive protons and negative
  electrons which are attracting and
  repelling with this great force. So
  perfect is the balance however, that
  when you stand near someone else you
  don't feel any force at all. If there
  were even a little bit of unbalance
  you would know it. If you were
  standing at arm's length from someone
  and each of you had one percent more
  electrons than protons, the repelling
  force would be incredible. How great?
  Enough to lift the Empire State
  building? No! To lift Mount Everest?
  No! The repulsion would be enough to
  lift a "weight" equal to that of the
  entire earth!

Another way to think about it is this: a proton has both charge and mass. If I hold another proton a centimeter away, how strong is the gravitational attraction? It's about $10^{-57}$ newtons. How strong is the electric repulsion? It's about $10^{-24}$ newtons. How much stronger is the electric force than the gravitational? We find that it's $10^{33}$ times stronger, as in 1,000,000,000,000,000,000,000,000,000,000,000 times more powerful!
A: This is indeed something one has to be careful about, because, after all, gravity scales with the mass of the particles in question, whereas the other forces scale with the electric charge or the magnetic moment. It appears that one compares apples with pears. 
However, I believe the declaration that gravity is the "weakest" of the forces stems indeed solely from its irrelevancy on the scale of particle physics.
A: Gravity is weak because the masses of elementary particles are so small.  Gravity has a natural mass unit, $m_p~=~\sqrt{\hbar c/G}$, the Planck mass, which is about $10^{-5}$ g.  The proton is $22$ orders of magnitude less massive.  So the stuff which makes up the world is elementary particle “styrofoam stuff” which gravity couples to.  
This can be seen as well with IIA strings and their S-dual heterotic strings.  Those heterotic strings just do not like to stay on our brane, which they have no end points to form Chan-Paton factors or Dirichlet boundary conditions on the brane with.  They slip through our brane as if nothing is there.  Their S-dual strings are open strings on the brane, but with puny masses --- far less than the Planck mass or the mass corresponding to the string tension.
A: The Randall-Sundrum model explains it. The other forces are confined to the brane which  we consider to be our universe. The brane is embedded in higher dimensional space where some of the dimensions may be compactified, but others could be larger or even infinite (a 5 dimensional anti-de Sitter space in which a (3+1)dim brane is embedded.All particles except the graviton are bound to the brane.) Higher dimensional space is called the bulk. If gravity is not confined to our brane and can penetrate into the bulk, that would explain its weakness. The problem with the extreme difference in strengths of the forces is termed the hierarchy problem (weak force=$10^{32}$grav force). There are other explanations involving supersymmetry.
A: When we say that gravity is much weaker then the other forces we mean that its coupling constant is much smaller than the coupling constants of other forces.
Think about a coupling constant as a parameter that says how much energy there will be in per "unit of interacting stuff". This is a very rough definition but it will serve our purpose. 
If you determine the coupling constants of all different forces, you discover that, in decreasing order, strong, eletromagnetic and weak forces are much, much stronger than gravity.
You need around $10^{32}$ (that is 100,000,000,000,000,000,000,000,000,000,000) times more "stuff interacting" to get around the same energy scale with gravity if you compare it with the weak force. Moreover, the difference between strong, weak and electromagnetic forces among themselves isn't nearly as extreme as the difference between gravity and the other forces.
A: To compare the strength of e.g. the electromagnetic force and the gravitational force one should not simply compare the difference of strength for a specific particle, because there are too many (electron, proton, myon, etc.) and all these comparisons would yield a different number. Instead let's compare gravity and EM force using a scale given by fundamental constants. For gravity this means, we choose the gravitational force between two particles of Planck mass $m_P=\sqrt{\hbar c/G}$ and for the EM force we compute the force between two particles of Planck charge $q_P=\sqrt{4\pi\epsilon_0 \hbar c}$. If we compute the forces for these two cases respectively we get
$$
F_g = G \frac{\hbar c/G}{r^2} = \frac{\hbar c}{r^2}\\
F_{EM} = \frac{1}{4\pi\epsilon_0} \frac{4\pi\epsilon_0 \hbar c }{r^2} = \frac{\hbar c}{r^2} 
$$
So we see the gravitational force and the electromagnetic force are actually of equal strength and range when they are compared at their natural scale. 
In this sense gravity is not the weakest force. Instead the puzzle here can be rephrased to: Why is the mass of the known particles so low compared to the Planck mass, or why is the charge of the known particles so high compared to the Planck charge?
For example the ratio between planck mass and electron mass (the proton mass should not be used for comparison, because a proton is not a point particle and it's mass is made complicated by effects of QCD) is
$$
\frac{m_P}{m_e} = 2.389 \cdot 10^{22}
$$
whereas the ratio between planck charge and the charge of the electron is
$$
\frac{q_P}{q_e} = 11.706
$$
The ratio of these ratios is $2.040 \cdot 10^{21}$ and why this is such a high number is the real puzzle, and I don't know any theory which could explain this number. 
It should be noted that the so called Mass-to-charge ratio for an electron is of order $10^{11}$, for a hypothetical particle with a Planck charge and a Planck mass it is accordingly of order $10^{-11}$ 
$$
\frac{q_e}{m_e} = 1.759 \cdot 10^{11} \frac{C}{kg} ,\ 
\frac{q_P}{m_P} = 8.617 \cdot 10^{-11} \frac{C}{kg} = \sqrt{\frac{4\pi\epsilon_0}{G}}
$$
which I think is a quite remarkable "symmetry".
UPDATE: The numbers shown here might not be familiar. But I want to note that the fine-structure constant is 
$$
\alpha^{-1} = \left(\frac{q_P}{q_e}\right)^2 = 11.706^2 = 137.036
$$
and the gravitational fine structure constant is
$$
\alpha_G^{-1} = \left(\frac{m_P}{m_e}\right)^2 = (2.389 \cdot 10^{22})^2 = 5.709 \cdot 10^{44}
$$
A: Not exactly an answer but a resource that bears relevance to this question: Richard Feynman had a few words to say about the notion in his messenger lectures series, lecture 1, around the 48:20 time mark
A: Gravity is the weakest force as its coupling constant is small in value.
Gravity cannot be felt by us in daily life because of the huge universe surrounding us. Electromagnetic force is undoubtedly stronger as it deals with microscopic particles (electrons, protons). Gravity is always attractive in nature. It is a long range force among all other interactions in nature.
