How is dark matter detected? What methods do we use to detect Dark Matter?
If I understand correctly, due to lack of electromagnetic interaction it should be able to phase through normal matter nearly like through void - since the structure of matter outside the (relatively tiny) nuclei is based upon electromagnetic interactions between atoms and within atoms. It also doesn't exhibit the Strong forces which bind the nuclei, making them somewhat "transparent" as well.
It acts gravitationally, but that's difficult to detect if there are no observable objects affected by the gravity there, and it exhibits the Weak interactions, which only very lightly affect nuclei of atoms.
So, what do dark matter detectors measure directly, deriving density of dark matter from the measurement? 
 A: As of now, there is no dark matter detector that directly measures dark matter densities. You point out the main problem: If dark matter doesn't interact at all except through gravitation, how can we measure it? I don't think we could. However, we don't know that dark matter doesn't interact at all, simply because we don't know what dark matter really is. It definitely doesn't interact much, but not at all? There are many proposals as to what dark matter will ultimately turn out to be and most of these proposals make predictions one can actually measure.
Hence, there are many examples for proposed measurements of dark matter such as Xenon1T which are based on various proposals of what dark matter actually is. Xenon1T for example supposes that dark matter consists of a new type of particles that do interact, just very very weakly (way more weakly than neutrinos. They are therefore called "weakly interacting particles" also called "WIMPs"). I discussed a few different approaches at directly measuring dark matter here. As of now, none of them has provided conclusive evidence for having measured "dark matter".
How do we know the density of dark matter then? Well, it's always indirectly. Take a galaxy: You can measure the rotation speed of objects in a galaxy and from the speed of the objects at various distances from the galactic centre, you can infer the mass density. Now you substract what you CAN see, i.e. usual baryonic mass and you notice that this is not enough. The rest is then just dubbed "dark matter", because we obviously can't see it and don't know what it is... 
And this is just one way. There are a many other areas where there must be more matter than people can "see" (see Wikipedia for more areas).
As pointed out in the comments, there was some flury of news about our solar system and even all planets having "hair", very dense filaments of dark matter. This idea (see this paper) however is not the result of direct measurements, but it follows from numerical and analytical calculations.
A: There is no direct way to measure dark matter. In SUSY and other theories, dark matter particles can interact with baryonic matter [wiki] https://en.wikipedia.org/wiki/Supersymmetry#Dark_matter) but this has no consequence at cosmological scales. The only way to measure dark matter is by analyzing the properties of baryonic matter and to see the velocity dispersions of galaxies by presuming that the baryonic matter is "observable". Dark matter particles can be cold, hot, or warm.
