The word "matter" needs to be used with a great deal of care. In science, the word has definitely passed its use-by date. Some problems with the precise usage of the word are listed on the Wikipedia page for Matter and I quote:
...As such, there is no single universally agreed scientific meaning of the word "matter". Scientifically, the term "mass" is well-defined, but "matter" is not.
Even mass is a word that needs to be used with great care. There are several, in-principle distinct meanings for it. Let's see whether we can untangle this mess.
For me and, I believe, most physicists, if one uses the word "matter" unqualified, it implies something with nonzero rest mass. Rest mass is the total energy of something that you measure when you are in an inertial motion frame that is at rest relative to the object in question (in SI units, you must divide the total rest energy by $c^2$ to get the rest mass).
Inertial mass is the constant in Newton's second law. It is the "stubborness" of the object in question: it measures how quickly or slothfully something reacts when you "shove" it with an impulse. It is always measured from an inertial frame that is at rest relative to the body just before the shove. Whilst it is true that for Newtonian physics any inertial frame will do, in special relativity this frame must be at rest relative to the body and the impulse infinitessimal to get the correct inertial mass.
Gravitational mass is the "coupling strength" of an object to the gravitational field (if you're doing Newtonian Gravity) - how much gravitational force the object exerts on its surroundings and how much it itself is acted on by gravity. In the Einsteinian picture, it is a "source" strength of an object: it's part of what you plug into the so called "stress energy" tensor and so can be thought of as a "source" to calculate the warping of spacetime around the object.
In turn, even the word "energy" which defines rest mass ultimately needs further qualification. In general relativity, it is the $0\,0$ component of the stress energy tensor: the "source" of spacetime curvature. Some people who study general relativity co-opt the word "matter" to mean "source of spacetime curvature", i.e. anything that begets a nonzero stress-energy tensor.
As far as we know, these three kinds of masses are the same. These equivalences are experimental facts (see for example the Eötvös experiment) and the gravitational / inertial equivalence is a beginning point for general relativity. A system with total energy at rest $m$ accelerates in response to a force $F$ at acceleration $F/m$. A system with total energy at rest $m$ experiences (in Newtonian gravity) a force $m\,g$ on it. The last equivalence, in particular, is tested constantly seeking tiny deviations from it in the hope of gleaning experimental data that would allow us to refine Einstein's theory of gravity. Look up, for example, the Nordtvelt effect. Or read up on the simple thought experiment by Paul Davies on quantum tunnelling in a gravitational field that suggests a deviation. The paper is cited in this Physics SE answer here
What are the modern usages? Physicists are most interested in states of the World where the quantum fields that make it up are in excited states. They have more precise names for these excitations, depending on which quantum field is involved: things like electron, photon, quark, muon .... and so forth. Unfortunately in general you need to specify exactly what it is that you're talking about. Also, mass by physicists is almost always taken to mean rest mass - this quantity is Lorentz invariant (independent of inertial observer). Sometimes you'll hear of relativistic mass - this notion has fallen into disfavor for teaching, but it refers to an object's increased inertial mass if it is moving relative to you. An impulse of $I$ will no longer beget a change of velocity $I/m_0$ when the body is initial moving relative to you (where $m_0$ is the rest mass) but rather the body behaves as though its inertial mass has increased to $\gamma\,m$, where $\gamma$ is the Lorentz factor.
As user Ruslan reminded me about the notion of relativistic mass:
Not quite $\gamma\,m$, rather it depends on angle between the force and velocity of the body. See discussion of longitudinal and transverse masses for details.