Here I will explain the difference between matter, anti matter, dark matter, and negative matter in a concise and understandable way.   I have seen confusion pop up in various online forums and comments on the recent announced trapping of an anti atom by CERN. 

The first thing to know is that for a physicist there are four fundamental forces of nature which are always at work.  These are the familiar Gravity and  Electromagnetism, as well as the generally unfamiliar strong and weak atomic forces.   The atomic forces work on the length scale of atoms, Electromagnetism and gravity work on the length scale of the universe though in fundamentally different ways.

The different kinds of matter interact through these forces in different ways. 

Normal matter like that which we are all made of interacts via all four forces in the ways which we are familiar with.  All of the science and technology we have done to this point is based on normal matter and how it behaves.  But for a few experiments and in certain particle accelerators all we have done has been with normal matter.   Hence it's name. 

Anti matter is just like normal matter only the sign of certain properties is different.  The classic case would be the electron, which has as it's anti particle the anti-electron also known as the positron.  Electrons are negatively charged, and Positrons are positively charged.  Yet they are identical in every other way.    Then their are particles like neutrons and protons which are made of even smaller particles called quarks.  Quarks interact via the strong atomic force, and electromagnetism.  Anti Quarks have opposite charges to Quarks in those two forces.

Dark matter on the other hand only interacts by way of gravity and the weak atomic force.  Dark matter does not interact via either the strong atomic force or electromagnetism hence dark matter cannot be seen and is hard to detect.  It only interacts via the weak force which is what keeps neutrons and protons inside the nucleus of atoms together.    Such is why experiments to detect dark matter directly rely on a particle of dark matter bumping into a particle of matter dead bang on the nucleus of an atom of normal matter.  

Most of the reason we think dark matter exists has to do with the fact that it solves problems in cosmology in a very expedient way without us having to alter General Relativity.  It is widely agreed that dark matter whatever it turns out to be quantifies how much we really don't know about the matter in the universe. 

Negative matter is a hypothetical type of matter which if it exist will have negative mass and negative energy.  It will in essence have a negative gravitational charge and repel normal matter.  Yet it will interact just like any other matter in every other way. 

This table summarizes the differences in how each type of matter would interact with the different forces.

 Normal Matter Anti- Dark- Negative- 
Gravity  As usual As usual* As usualOpposite sign 
Electromagnetism As usual Opposite sign No charge ?
Strong Force As usual Opposite sign No charge ?
Weak Force As usual Opposite sign As usual ?

*We assume that antimatter behaves as normal matter under gravity.  The truth is we have never seen a large enough mass of it to know for certain it behaves the same.  When it comes to Negative matter we know nothing and it may not even exist outside of certain theories.  Dark matter is on the edge of being a confirmed real entity.