Why don't they use electrons at CERN?

The choice of particles for a collider depends on what needs to be clarified next in the zoo of particle physics and the theory of the Standard model, and also depends on the difficulties introduced by the particular particles used. Here are the current accelerators at CERN.

Creating a beam means to accelerate charged particles, whether electrons, protons, positrons or antiprotons, and accelerating particles radiate energy away as photons. At relativistic energies it is called synchrotron radiation. Electrons radiate away a lot more energy in order to reach the same beam energy as protons, in the circular accelerator case , for each turn the power,P, lost in radiation is given by


where $ρ$ is the radius, and $γ$ in terms of the beam energy $E$, is given as $γ=E/m_0·c^2$. So the larger the mass and radius the smaller the power lost. In the linear accelerator case the power lost depends less drastically on mass, see this for synchrotron radiation.

So it is much more expensive to bring a beam of electrons to a given energy than a beam of protons to that energy. For exploring new regions of mass, proton beams are better than electron ones in terms of cost of experiment.

Proton on antiproton beams were used in the SPS at CERN to explore high energies. That is how the W and Z bosons were discovered. The next accelerator was an electron positron accelerator, LEP, specifically designed to study in detail the interactions of Z and W. Electron-electron scattering is not very efficient in giving possible productions that could be described well theoretically. Electrons on positrons, as elementary particles, give much cleaner events and possibilities for calculating.

After LEP, which almost had reached the now known Higgs mass, the same tunnel was used because protons on protons can have much higher energy then electrons and positrons in the same circular tunnel, an economy on the expense, and again LHC is an exploration machine.

The next machine that the high energy international community aims at is a linear collider of electrons on positrons, to avoid the large loss of energy by the leptons in a circular collider at the energies aimed. Again the leptons are chosen so as to have a clean vertex of point elementary particles in the calculations of the interactions.

So electrons were used and will be used in the future accelerators, in conjunction with positrons, to explore the energy regime further. Electrons on protons would still have the ambiguities introduced by the complexity of quarks, although there are experiments studying specifically electron proton scattering, to study the proton. LHC was aimed at studying the highest energy possible interactions of matter, at the time it was built, for exploratory reasons. The ILC will be exploring the fine details of that highest energy range of 500 GeV to 1TeV.

I would like to add to Anna_v 's good answer some details which seem to me important. Collider physics already reached some maturity, so one can recognise now the role of different collider types on the evolution of particle physics.

Proton-colliders (respectively proton-anti-proton colliders) serve typically for the discovery of new fundamental particles, whereas electron, better electron-positron colliders, serve for a precision measurements of the laws of particle physics, i.e. the standard model. Therefore to a large extent LEP (electron-collider) for instance provided the experimental confirmation of the standard model.

However, both types of colliders have their limitations. The main limitation of electron-positron colliders is the generation of a tremendous amount of synchrotron radiation that goes like $\sim \gamma^4$ ($\gamma$ is the energy-(mass$\times c^2)$ ratio of a particle). So trying to increase the energy further is difficult to achieve as the aspired energy increase mostly gets lost in synchrotron radiation. Electrons/positrons can easily reach very high $\gamma = 10^5$ for 50GeV electrons, a typical energy of electrons/positrons in the LEP-ring. On the other hand for the same energy the proton's $\gamma = 53$, so the synchrotron radiation production is much smaller. So even at 7TeV proton energy (LHC) the $\gamma = 7463$ which is still smaller than the one for 50GeV electrons. On the other hand it is only one parton (one valence or sea quark or gluon) in the proton that will effectively participate in the collision. Therefore only a part of the energy provided by the collider to the proton will be used for the collision. That is not the case for electron or positrons. The other constituents of the proton are kind of spectators and will after a process called hadron-fragmentation end up in a swarm of hadrons that are of little interest and increase significantly the background to the measurement due to their presence in the detector. The energy of the electron/positrons, however, will be fully used in the collision and the collision result can be observed in the detector with only very little background, in other words the result is clean. Therefore it is so useful for precision measurements.

Proton colliders have one of their limitations in the difficult detection of the numerous particles that are produced in the collision. But up to now the experimental particle physicists were clever enough to find the new particles they were looking for: Examples: $Z^0, W^\pm$, t(op) and H(iggs) were all discovered at proton colliders. The LEP (Large-Electron-positron-collider) strived to find the top-quark and the Higgs, it was not able to find these particles as they were heavier than expected (increasing the energy of the electrons further was not a viable solution among other reasons due to the high energy loss due to synchrotron radiation.).

In the meantime, collider evolution progresses in alternance: SPS(proton-collider), electron-collider (LEP), proton-collider (LHC), and probably the next generation will be the FCC-ee (Future Circular-Collider with electrons) followed by the hadron/protron version FCC-hh. The FCC-ee will have circumference of 97km compared to the 27km of LEP/LHC, so the amount of synchrotron radiation, in particular from electrons/positrons, will be reduced partially by a larger circumference. The FCC-ee will serve for the precision measurement of the Higgs and its unusual potential, in order to check if the particle found 2012 is indeed a standard model Higgs (or in order to explain the measurements new physics, for instance Supersymmetry, is necessary). The FCC-hh will again try to explore new energy ranges in order to find new particles as it does now the LHC. In Europe now all the efforts go in the development of the FCC (-ee or -hh), it seems that other linear collider projects using electron/positrons as ILC (Japan) and CLIC (CERN) are no longer followed up seriously. I think, one of the reasons why a FCC is favoured is that once the tunnel is excavated it will be used first for an electron-collider followed by a protron-collider. So it is the same with LEP and the LHC: Both were built in the same tunnel.

So your question why not using an electron-collider is justified, but each collider type has its own era.