I can certainly try!
In quantum field theory we model two particles interacting via a force as one of those particles emitting a particle and the other absorbing it (well, it can get more complicated but that's the basic principle). The particle that is emitted/absorbed is said to "mediate" the interaction and each force is mediated by a different particle/set of particles. You can think of it sort of like two people standing on boats, if you throw a ball to the other boat then to conserve momentum your boat will move opposite the direction the ball was thrown and the receiver will catch the ball and move slightly in the direction the ball was moving. So the overall effect is that you are both pushed away from the other.
In the standard model electromagnetism is mediated by the photon so any two electrically charge particles will interact by the exchange of photons. Weakly charged particles interact via exchange of the W and Z boson and strongly charged particles interact via exchange of gluons.
As for the Higgs, this is a bit more complicated since it's really rooted in the mathematics of QFT. Briefly put, without the Higgs boson there are two sets mathematical transformations associated with the Weak and Hypercharge force respectively. These are mathematical transformations which change the maths behind the scenes but don't change the physical predictions in anyway (see http://www.reddit.com/r/AskPhysics/comments/2whdio/can_anyone_do_an_eli5_of_gauge_theory/ for good explanations of this). The weak force is associated with a symmetry (set of transformations) we call SU(2) and the hypercharge with a set called U(1). Together these give an overall symmetry SU(2) x U(1).
It is worth mentioning that in physics a symmetry is always associated with some conserved quantity such as a charge, in addition, the types of symmetries we are talking about here (local gauge symmetries) also be necessity give rise to particles (the force carrying bosons) so each symmetry group gives you a force with it's own charge and set of particles. A U(1) symmetry gives you 1 particle and a SU(N) symmetry gives you N2 - 1 particles.
However, if we introduce the Higgs boson then these symmetries are no longer valid, individual SU(2) or U(1) transformations no longer give correct physics because the Higgs doesnt transform correctly. If you work through the maths then you find the two W bosons carry on as is but gain mass, the A and B bosons never appear on their own though, instead they always appear with mathematical terms of the form (A + iB) or (A - iB). If they never appear seperately this is absolutely physically indistinguishable from having two particles (say the Z and photon) where Z = A + iB and photon = A - iB (or vice-versa I can't remember of the top of my head :) ).
Once you have done all this you find that the only remaining symmetry is a U(1) symmetry like hypercharge but that the charge associated with it is not the hypercharge but a combination of hypercharge and weak charge (literally for each particle with weak charge T and hypercharge Y the electric charge Q = T + Y/2). This is exactly the electromagnetic force.
As I said, it's a bit complicated! I can try and explain further if anything's unclear.