How can we detect particles that have no electric charge? Particle accelerators look for electric changes, but what if particle has no effect on other particles or an electric charge, would we not detect it?
 A: There are 3 neutral particles which are stable on  the time-scale of collider events: $\gamma$, $n$,  $\nu$/$\bar{\nu}$.
(Anti) neutrinos (mostly from meson, muon, and $Z$ decays) cannot be detected, and escape. In events like:
$$X \rightarrow \mu + \nu_{\mu} $$
the neutrino can be inferred from missing (transverse) momentum. In a $pp$ collision in which two partons interact, the initial state longitudinal momentum is not known, but the transverse momentum is zero. A muon will be detected with large transverse momentum, leaving "missing momentum" that is attributed to the neutrino.
Neutrons are basically a non-factor in colliders. Specific medium energy nuclear physics experiments look at final state neutrons, usually with a sensitivity to the neutron polarization. See http://galileo.phys.virginia.edu/Research/groups/INPP/gen/ndet/ndethome.html , for instance.
Photon detection is an important part of collider experiments, as many processes of interest produce hard $\gamma$'s. For this reason, the outer layers of detectors have lead-glass calorimeters. The neutral photons will traverse the lower-Z detector material and interact with the lead in the $Pb$-glass via pair-production in the strong electric field near the nucleus:
$$ \gamma \rightarrow e^+e^- $$
followed by Bremsstrahlung:
$$ e \rightarrow e\gamma $$
and so on in an electromagnetic shower. Energy loss occurs via $dE/dx$ ionization which is roughly constant across observable energies. Hence: the total charged particle track length is proportional to $E_{\gamma}$.
Since the lead is in transparent glass, the $e^{\pm}$'s emit Cherenkov light. The total amount of light emitted is proportional to the track length, and hence: $E_{\gamma}$.
Meanwhile, muons with their higher mass do not initiate showers. They traverse the calorimeter and can be detected with scintillator paddles on their way out.
A: 
Particle accelerators look for electric changes, but what if particle has no effect on other particles or an electric charge, would we not detect it?

The effects are called interactions, there there are four interaction forces. Gravity we can ignore at the particle level, but there is the electromagnetic, the weak, and the strong, each described with its own strength. How probable it is that a particle hitting another particle will interact depends on the strength  the couplings .

Take the LHC, when protons hit on protons a large number of particles are created with the energy supplied by the collider, primarily due to the strong interaction,the product particles may be charged or neutral, but they all have energy. The huge detectors are designed so as to get with continuing interactions the type of particles.
It is only neutral particles that interact only with the weak interaction that can pass through the detecting system unmeasured, and they are "detected" by energy and momentum conservation considerations. The neutrinos are in this category, but also searches for new physics may be looking at missing energy and momentum as a signal of something new.
Take the cms detector:

The figure of the man gives a feeling for the size, and the complexity is evident. That is why it takes thousands of people from all over the world to build, maintain , and analyze such experiments. How the particles are detected is analyzed in the link. The Higgs CMS paper was authored by over 5000 people.
Here one can see a list of real events measured with the detector.
In this answer of mine to a related question I discuss how the detectors in current LHC experiments work. Also here .
