Why doesn't the LHC accelerate electrons?
Whenever you accelerate a charged particle it emits EM radiation known as Bremsstrahlung, and obviously charged particles moving in a circle are accelerating (towards the centre). This means that any circular collider emits a continual stream of Bremsstrahlung radiation. To counteract the energy lost to Bremsstrahlung you have to put energy in, and that costs money and annoys the local power companies.
For a given beam energy the Bremsstrahlung losses increase with decreasing particle mass, so it costs a lot more to run an electron collider than to run a proton collider of the same energy and beam current. The LEP collider, with a maximum energy of about 200GeV consumed around 70MW when running, while the LHC with a vastly higher beam energy only consumes around 120MW. These figures are a bit misleading since they include the costs of cooling, etc, and not just running the beam. According to this article the power required for maintaining the beam at the LHC is only around 20MW. I haven't been able to find the corresponding information for LEP.
All the proposed future electron/positron colliders are linear. This avoids the Bremsstrahlung losses when you bend the particle beam.
The energy lost by a particle doing one turn in a circular machine is
$$U_0\propto E^4R^{−1}m^{-4}$$
where $E$ is the beam energy, $R$ is the bending radius, $m$ is the mass of the particle that you want to accelerate.
It comes out that for the mass of heavy particles such as muons, protons and heavy ions, the field strength of the bending magnets is still the limiting factor, but light particles such as electrons and positrons are simply radiating too much energy.
The problem is not simply wasting energy, but how to push that energy back to the beam. The acceleration is normally done in straight sections of the ring by using a radio-frequency electric field. If the field is not strong enough (in relation to the available space) to compensate the energy lost in the bending sections, the machine will never work even with an endless amount of energy available from the grid.
So you are forced to increase the radius to reduce the energy loss and have more space to compensate it, but looking at the relation above, you see that you cannot go too far. For instance TLEP plans to have a radius tree times larger than LEP, but its energy won't double the LEP one and it might be built only in view of a much more energetic proton machine coming in future in the same tunnel.
The other way is to build straight colliders such as CLIC or ILC projects, which however do not come without technical difficulties.
Apart from the reason mentioned in previous answers (Bremsstrahlung) there is one more thing why proton collider is used: it can scan wide range of collision energies.
Because protons are compound particles, their collisions are in fact collisions of the quarks or gluons. These constituents have random energies and thus each collision typically has a different energy. This is handy when you look for a particle with unknown mass (like the Higgs boson). So you smash protons at 1.4 TeV, but in fact the actual collisions are in wide range of energies from MeV to TeV. This way some of the collisions will be at the level required to create the unknown particle. And you collect a lot of events to capture significant statistics to detect the particle.
On the other hand: lepton colliders collide leptons always with the working energy. You cannot explore energies outside the design range of the lepton collider. Thus lepton accelerator is handy when you want to focus on research of a single particle with well known mass. You tune the energy to the mass of the particle and get the data much faster. But you have to know the particle mass upfront, which was not the case for the Higgs.