The weak force can be better understood when taught together with the electromagnetic force. The resulting force is called electroweak.
First of all, consider electrons interacting with photons. The photons are symmetric under U(1) gauge group. Then consider neutrinos. They interact much like photons by exchanging Z bosons. This is another U(1) subgroup. Now - can electrons interact with neutrinos? They can, and this is done by W bosons. Electron has electric charge and neutrino does not, so the mediating boson must have charge itself (to carry it from one particle to another). The electron-neutrino case is also not symmetrical under permutation of the particles, so the mediating boson is not its own antiparticle. We call them W+ and W-.
When you write it down, you get a 2×2 table - the rows and columns are indexed by particles (electron, neutrino) and each cell holds the boson that carries interaction between them. This 2×2 matrix is together symmetrical under SU(2) gauge group. That means, instead of pure electrons and neutrinos, you can take their quantum superpositions in certain linear combinations - and nothing will change. This is a mathematical operation similar to a rotation. You can imagine an abstract vector that gets rotated in some abstract space. This vector has been named the weak isospin. You can treat the W bosons as rotations 90 degrees left or right. Photons and Z bosons don't have such nice visualization, but they could have. You can rewrite your matrix of bosons in another (but equivalent) representation by taking various linear combinations of photon, Z boson and W bosons. The most useful base consists of four bosons: B, W1, W2 and W3. These bosons are quantum superpositions of our previous bosons, with certain factors. They have less physical meaning, but they have nicer mathematical properties. In particular, you can look at them just like at the rotations. The B boson would be a unit matrix. W1, W2 and W3 would be rotations aroud three "axes". These axes are quite abstract and they mean certain quantum superpositions of electrons and neutrinos.
Actually, there is one more U(1) symmetry in this picture and it's called weak hypercharge. The B boson is the generator of it (despite it is a unit matrix for isospin). The full symmetry group of the electroweak interaction is SU(2)×U(1).
The biggest challenge with this approach is the obvious fact that photon, Z and W bosons have very different masses. Why do we happily substitute one boson by another, when they are so different particles? This question gave life to the one of the most beautiful theories in physics.
Imagine again our space of isospin. We said everything was symmetric under it. What if it wasn't? Let's introduce some field that sets some direction in the space of the isospin. I like to visualise it as a wind flowing through the isospin space. We have a preferred direction (the wind direction). If we take a sheet of paper and put it parallel to the direction of wind, it will not be affected. If we put it at any other angle, it will be. If we put it perpendicular to the direction of wind, it will be affected with maximal strength. This is called symmetry breaking. Now we can explain the difference between Z boson and photon. Photon is like a sheet parallel to the wind. It does not feel the symmetry breaking field. On the other hand, Z boson feels it with maximal strength. That's why they have different masses. The symmetry breaking field is of course the famous Higgs field. Interaction with this field gives masses to gauge bosons, the more the better "align" of the boson direction in the isospin space with the Higgs direction.
The electromagnetic interation is in fact a "line" (subgroup) in the isospin and hypercharge "space" (group) that is unaffected by Higgs. That's why photon is massless and the electromagnetic interaction is the strongest. In fact you can have infinite number of different "electric charges" and "interactions", but they all are weaker than the ordinary electromagnetism. The weakest of them (most prone to Higgs) is the attracting force between neutrinos with Z bosons as carriers. That's why we call it the weak interaction.
What is the angle of the Higgs field "wind" compared to the direction set by the B boson and one of the W bosons? Its numerical value is known from the experiment, but it has no explanation. It is called the Weinberg angle. It is very close to exactly 30 degrees. Maybe it's some law of physics.
That's how theory postulates SU(2)×U(1) symmetry despite no visible symmetry at all. This symmetry would be visible if the Higgs field was somehow turned off. There are theories that the cosmological inflation was in fact a period when Higgs was turned off and particles didn't have mass. In such conditions, according to general relativity, the spacetime expands exponentially, just like inflation theory predicts. This is called the de Sitter solution of the Einstein equation.
To get the full picture, you must include quarks. Actually up and down quarks also set two orthogonal directions in the isospin space, just like the electron and neutrino. Their electric charge difference is also one elementary charge. They also can transform one into another by exchanging a W boson. They also attract themselves with photons (and, to lesser extent, Z bosons). If not for the color (strong) interaction, up and down quark pair would closely resemble electron and neutrino pair.
The isospin space is similar to the normal spin space and different particles may have different mathematical properties under it. We could have isospin scalars (particles not interacting weakly), spinors like electron and neutrino pair or up and down quark pair, vectors like γ, Z, W, B bosons and so on. Compound particles have isospin equal to the sum of the isospins of their components. Neutron and proton are their respective isospin rotation images. Also atomic nuclei with equal number of nucleons are connected this way. Say, tritium, helium 3 and lithium 3. The name "isospin" stands for "isotopic spin", because it was first discovered in the properties of such isotopes.
The hardest thing to understand with the weak interaction is that it is chiral. That means, only particles with certain chiralilty are components of non-scalars in isospin space. If you reflect an electron in a mirror, it will not undergo weak interactions. It will be a scalar. That's just a mathematical formulation of the fact. Our universe is not symmetrical under mirror reflections. We don't know why.
