I'll try my best to explain the Higgs mechanism without any math. It goes very roughly like this:

1. All elementary particles are merely excited states (or quanta) of some field. This includes the Higgs boson, which is the quanta of the Higgs field, the photon, which is the quanta of the electromagnetic field, the electron, which is the quanta of the electron field, and so on. All fields exist at all points in time and space.

2. Fields may couple to other fields, and in this case the fields are said to be interacting with one another.

3. Some fields couple to the Higgs field. After a process called spontaneous symmetry breaking, the Higgs field is separated into two parts. The first part remains a dynamic field, and its quanta are the Higgs bosons. The second part is a constant (called the vacuum expectation value), and the equations that describe the coupling of the Higgs field to other fields become equations that describe the other fields coupling (quadratically) to themselves, which in quantum field theory is interpreted as giving mass to a field. The vacuum expectation value of the Higgs field is therefore proportional to the mass of each field. (See further explanation below.)

4. The equations that are interpreted as giving mass to certain fields do not exist before the spontaneous symmetry breaking of the Higgs field occurs. Actually, they cannot exist, due to symmetry considerations (and this is why it's called "symmetry breaking"!). So this is how the Higgs field gives masses to the elementary particles: any field that couples (or interacts) with the Higgs field acquires a mass term that would otherwise would not have existed.

5. Going back to item (1), since all elementary particles are quanta of their corresponding fields, the particles that are the quanta of fields that couple to the Higgs field acquire mass due to spontaneous symmetry breaking, which is the essence of the Higgs mechanism. This includes all known particles (or fields) except the photon, the gluon, and (possibly) the 3 generations of neutrinos.

EDIT: As requested in the comments, I will attempt to explain a few things.

Regarding item (3) above, what exactly is the vacuum expectation value?

(Note: I took the illustrations from a post on the blog Quantum Diaries, which you are encouraged to read.)

The vacuum expectation value of the Higgs field is just the value that we would "expect" it to have when it is in its vacuum state, which is the state of lowest energy. It turns out that it is a general law of nature that physical systems always "want" to be in the state of lowest possible energy. The allowed values for the energy are determined by the system's potential energy function. In the case of the Higgs field, the potential function looks (more or less) like this:

This is called the "mexican hat" potential for obvious reasons. It's a 3-dimensional graph. The two horizontal axes are the values that the field can take. The vertical axis, labeled [math]V(\phi)[/math], is the value of the potential energy that corresponds to each specific value of the field [math]\phi[/math]. It's kind of like a geographic terrain, where the values of the field are the longitude and latitude, and the value of the potential is the height.

So we have a large valley, with a small hill at the center, on which the Higgs field is currently "sitting" in the image. However, the Higgs looks around and notices that there are lower energy states all around him, at the bottom of the valley. It "wants" to roll down the hill into a state of lower energy.

Notice that when it's at the top of the hill, the system is completely symmetric; you can rotate the potential around the vertical axis as much as you want, and it'll still look exactly the same. But after the Higgs rolls down into a particular spot, the potential is no longer symmetric. We call this process spontaneous symmetry breaking, because the Higgs "broke" the symmetry spontaneously when it chose a specific point on the circle to roll down into. Here is an illustration of what happens:

The Higgs chose a particular point to roll down into, on the right of the hill, and if we now rotate the potential function in the direction of the blue arrow, it will no longer be at the same point. So the symmetry was broken.

When the Higgs rolled down into a point of lower energy ("height"), we say that it acquired a vacuum expectation value (VEV). Note that the VEV is the value of the field, not of the energy. Previously, when it was at the top of the hill, the field's VEV was zero; this can be easily seen from the fact that it was at the origin (center) of the horizontal plane, where the field equals zero. Now the field has a non-zero value, the VEV, but it has lower energy than it had before.

Ok, those are nice pictures and all, but how does this process actually give mass to particles? I'm afraid for that you'll have to endure a little bit of math, but I promise it'll be really simple.

As I explained in point (3) above, some other fields couple to the Higgs field. This means that, in the equations that describe all the fields, there are some interaction terms that look (very roughly) like this:

[math]g\overline{\psi}\psi\phi[/math]

Here's what each symbol means:

• [math]\phi[/math] is the Higgs field.
• [math]\psi[/math] and [math]\overline{\psi}[/math] are the fields of some particle and its antiparticle. For example, an electron and a positron.
  
• [math]g[/math] is just a number, called the coupling constant, which determines how strong the interaction is between the three fields (electron, positron and Higgs).

Now, as described above, the Higgs field obtains a VEV. This is just some number. Let's call that number [math]v[/math]. So we can separate [math]\phi[/math] into [math]v[/math], which is just a number, and [math]H[/math], which is a new field:

[math]\phi=v+H[/math]

Let's put this into the expression above and see what we get:

[math]g\overline{\psi}\psi\phi=g\overline{\psi}\psi v+g\overline{\psi}\psi H[/math]

The expression on the right is still an interaction term, since it still has three fields. We just replaced [math]\phi[/math] with another field, [math]H[/math]. This new field, and not [math]\phi[/math], is the Higgs boson. So we got a term that describes how the electron and positron interact with the Higgs boson. But that's not relevant right now.

The expression on the left is where the mass comes from. First, let's combine [math]v[/math] and [math]g[/math] together, since they are both just numbers. And let's call that combination [math]m[/math]. So we have [math]m=gv[/math], and the expression becomes:
[math]m\overline{\psi}\psi[/math]
This is an "interaction term" between a particle and its antiparticle, and there is no third field. Such an interaction is called (drumroll...) a mass term! So, according to quantum field theory, this term says that the electron and positron both have mass [math]m[/math]. They didn't have it before; there was no mass term before. But with the help of the Higgs field's VEV, we've managed to create a mass term "out of nothing". This is how the Higgs field gives mass to particles.

What about particles, like protons, that do not acquire mass through the Higgs mechanism?

The Higgs mechanism can only give mass to elementary particles. The number of elementary particles is actually quite small. Here is a table of all the elementary particles and their properties:

There are 17 particles in this table, including the Higgs boson itself. Out of them, only 12 particles get masses from the Higgs mechanism. These are the 6 quarks u, d, c, s, t, b, the 3 leptons e, μ, τ, the 2 gauge bosons Z, W, and the Higgs boson itself, H. (It's possible that the neutrinos also get their masses from the Higgs mechanism, but we're not sure yet.)

However, there are many other particles that are not elementary; they are called composite particles. These particles are made from elementary particles and/or from other composite particles. For example, the proton is made from two u quarks and one d quark:

However, the proton's mass is around 940 MeV, which, as you can see from the table above, is a lot more than the sum of the masses of two u quarks and one d quark, which is around 9.4 MeV - only 1% of the proton's total mass! How is this possible? Well, we all know that [math]E=mc^2[/math]; energy is equivalent to mass, and vice versa. So the rest of the proton's mass must come from the energy stored within it.

Indeed, there are two sources of energy inside the proton. The quarks always move around inside, so they have kinetic energy. And the quarks also interact with each other (as illustrated by the squiggly lines connecting them); this interaction is what binds the quarks together, and it also has energy. So both the kinetic energy and the binding energy of the quarks contribute to the overall mass of the proton. The same goes for all other composite particles.