No, not all matter has mass. Most elementary particles have no mass. Photons do not have mass. These things are still called "matter", however (this is disputable, see below). "Mass" is simply that which is the constant of proportionality between force and acceleration, that is to say, when equal force is applied to a more massive body, it will accelerate more than the same amount of force applied to a less massive body , roughly corresponding to "how much" of an object there is, which is not to be confused with molarity, which is a measure of "what number" of objects there is.

Many people are confused and think that matter and energy are the same thing. THey aren't. Mass and energy are the same thing. In relativity, even inertially massless particles have mass because they are never at rest. This means, for example, that an input of energy can increase mass, but in everyday life this is irrelevant since even a single gram of mass is equivalent to over 10e^11J. "Energy" simply constitutes a scalar physical quantity used to do work on bodies, and all things have an associated energy quantity, as a scalar conserved property, constrasted to something like momentum, which is a vector conserved property. As for "matter" the notion is virtually impossible to define. Some people only consider massive bodies to be "matter", therefore excluding things like photons. Other scientists define it as "anything that takes up space". It's not clear and the debate does not show signs of resolution.