I believe only one person has mentioned what mass historically
signifies. Mass is a measure of the quantity of material, which roughly
translates into the number of protons and neutrons in the sample being
massed.
Most responders in this thread have been bringing up gravitational mass
and inertial mass. This is not surprising because gravitational forces
and inertial forces are how the quantity of material (mass) typically
manifests itself to us. Moreover, physics educators spend a lot of time
trying to teach about forces and acceleration and about gravitation.
However, defining mass through inertial or gravitational examples does
not strike me as correct. Rather, inertial and gravitational concepts
are ways to measure mass as opposed to ways of defining mass.
The mass of an object is a measure of the quantity of matter in the
object. You can decide how detailed to get about the definition of
matter, i.e. atoms, or neutrons and protons, or baryons and leptons
minus binding energies, etc.
That the quantity of material in a sample (the sample's mass) affects
the force of attraction between that sample and other samples is not
something most people find intuitive because most people don't have any
concept of how gravity works, i.e. how action at a distance works.
However, if one of the samples is Earth then people do have intuitive
knowledge that gravity exists and things have weight and a sample's
weight can be used to measure how much of that sample is present.
Therefore knowledge that an object's weight is a way to measure mass
(quantity of material) is fairly intuitive for most people. Therefore
the idea that mass is what you measure on a balance is not bad in terms
of intuitive feel, but I don't think it is a good definition of mass.
Rather, it is how you compare masses.
That the quantity of material in a sample affects how easy it is to
accelerate the sample is also something that is not real intuitive
although most people have a little bit of feel for it. The problem is
that their feel for inertial mass is tainted by lack of understanding of
acceleration and also tainted by friction. We physicists know that an
inertial balance can be used to measure mass and that a properly working
inertial balance gives the same numerical result as a properly working
equal-arm balance. But most people don't know that.
In either balance, gravitational or inertial, we are making mass
comparisons (to a standard) as opposed to making a mass definition.
I believe textbooks, particularly in chemistry, are beginning to teach
that mass is quantity of material. I think this is good because it is
correct and it is something the typical person can understand.
I was looking at some NIST resources on this, and I found an interesting
paper that some of you will also find interesting. I found it
particularly interesting to find out how the standard masses are
cleaned, how they are compared, how far off they are from each other,
how they are changing, and how they are divided into smaller amounts.
The paper can be found at...