Non-baryonic dark matter is the dominant matter component in the universe. As such its properties are directly linked to the evolution of the universe. All matter consists of particles. The particles that constitute the dark matter must have properties such that they lead to a universe as we observe it. Thus we can use observations of the universe, specifically its structure on large scales (cluster of galaxies and beyond), to learn about the properties of dark matter particles.
The total abundance of dark matter in the universe is another important piece of information: from the abundance of the particles we can learn something about how they were produced in the Big Bang, which in turn is related to their possible interactions with ordinary matter.
We know that dark matter is part of our own galaxy. However, so far we have not yet observed any direct effects from dark matter particles interacting with ordinary matter. This tells us that their interaction probability is rather low.
In the previous section we have learned that the particle properties of dark matter are closely related to the structure of the universe. If particles are produced in the Big Bang, their velocity is closely related to their mass: the lighter they are the faster move. Further, we observe that the universe on large scales is not filled uniformly with matter; instead we have large empty areas (voids) separated by narrow filaments and clumps of matter. This clumpy structure requires that the dominating species of particles was moving slowly at the time the structure formed. This hints towards a fairly massive dark matter particle.
We also learned in the previous section that dark matter particles must be interacting with ordinary matter only weakly (i.e.: the probability that such a particle will interact when it passes by some ordinary matter is very low) so that they could avoid being discovered until today. Also the total abundance points to a fairly weak interaction. These are some of the reasons why Weakly Interacting Massive Particles (WIMPs) are considered one of the most attractive classes of particles to solve the dark matter problem.
This hypothesis is also supported by modern particle physics: we know that our present picture of elementary particles and the forces acting between them, the so called Standard Model, is incomplete. Many of the proposed extensions of this model include new (i.e. so far un-observed) elementary particles, which are heavy and interact only weakly with the known particles. These new models have mostly been developed purely from a particle physics point of view and not with the dark matter problem in mind.
The fact that we find this coincidence of particle properties between the cosmological and astrophysical observations of dark matter on one side and particle physics on the other side is a very compelling motivation to look for direct evidence for WIMPs.
Other possible candidates are more exotic objects, like small primordial black holes (primordial meaning that they have been produced before atomic nuclei have formed in the early universe, while black holes we usually talk about are produced from collapsing heavy stars - much later in the history of the universe), or even the gravitational force that comes to us from a universe outside our own.
At this point we cannot decide which of the discussed candidates (if any...) makes up the dark matter in our universe, but WIMPs seem for several reasons to be one of the most straight forward explanations and we have a clear idea what we have to do to find direct evidence if they indeed are the solution to the dark matter problem.