First of all
- Normal matter ~ 5%
- Dark Matter ~ 20%
- Dark Energy ~75%
of the universe. I'm not correcting you, because you said 70% of all matter, just wanted to write this so there is no confusion.
Now, dark energy is very strange and we do not have any candidates, so we'll pass it from now.
Dark matter comes in two kinds: the first one is basically everything but stars (black holes, planets, dark galaxies etc.) Difficult to see, but made of atoms and just like the stuff we have here. This is the baryonic dark matter, which makes for a small fraction of the total amount. The second kind is not made of atoms, which makes it more interesting. It is further subdivided into particles that behave ultrarelativistically, relativistically and non-relativistically. Possible candidates are neutrinos, axions and WIMPs.
As to the question of why do we not feel it here, we don't know yet... The different kinds of dark matter are all needed for several reasons (cosmological, structural, etc.) but until we know what kind of particles compose it and how were they created we cannot explain their apparent abscence on Earth.
Take neutrinos, for example. A neutrino can pass through a one light year long block of lead with a 50% probability of being undisturbed. We are continously being pierced by millions of them and we do not feel a thing. They do not interact via the electromagnetic or strong nuclear force, which amkes them very difficult to detect. The experiments currently trying to measure the neutrinos coming from the Sun are incredibly complex and big. What I mean by this is that Dark Matter, even its strangest components, may very well be present here, even though we do not detect it.