Seeing how the other answers mention things like Lagrangian and quite a few details of the Standard Model, I feel that a more popular level explanation is needed. This comes with the pitfall of not being accurate, but for that you have other sources. I will pull a rabbit out of my hat and answer without a single (real) formula.
What does "mass" mean to a particle physicist? How does it relate-to/explain the classical notion of mass, and how does it differ?
Without going into definitions (you've received a link in you comments section), it is important to distinguish between two notions when talking about mass.
- Rest mass: this is the mass of an object at rest, that is, with 0 velocity.
- Measured mass (energy): this is the mass of an object which will be measured. It is composed of the rest mass plus a contribution we'll call "motion mass" produced by the non-zero velocity of the object.
There is some caveat relating to frames of reference in relativistic theory, but we'll skip that.
In classical physics, objects don't move fast enough to cause a distinction. Here is a graph to give you the idea of the change of measured mass with velocity (as a fraction of the speed of light):
When we talk about mass in particle physics, we mean the rest mass. Because the measured mass depends on the current (possibly changing) velocity, it is of little use as a defining property.
A note about "massless" particles
When you read/hear about a "massless" particle, it means that its rest mass is 0. You can now ask "Since the measured mass is comprised of the rest mass and the 'motion mass', and the rest mass is 0, what happens if the 'motion mass' is also 0?" Turns out particles with 0 rest mass behave a bit differently and must move at the speed of light. A photon's (a "particle of light") measured mass will be dependent on its frequency instead. Yes, light has a measured mass. You might be surprised because you never feel it, but look into solar sails to see that light can push other objects.
in the context of particle physics you hear things like "computing the mass"
Unfortunately, you can't put a particle on a scale and read its mass like you do with potatoes in the supermarket. A basic idea in computing the mass of a particle is using the law of energy-momentum conservation ("conservation of matter and energy", A.K.A. "law of equivalent exchange"). The law basically states that what comes in must come out.
By perform enough experiments in a particle collider, you will have enough data of what comes in and what comes out (you measure the measured mass) and will be able to compute the rest mass.
"such and such interaction gives some mass"
This and all the Higgs-related part is a bit more difficult to explain because it mostly arises from mathematics. I wouldn't go too much into the bouncing stuff in the video. I will try to approach this from the view point of interactions.
Not all particles can interact with the rest. For example, the aforementioned photon interacts with particles if and only if they have a (non-zero) electric charge. For the Higgs, you can make a similar argument only with mass instead of electric charge. Please note that the Higgs is not the equivalent of the photon when it comes to mass - see the graviton and this question. However, you can say that a particle has mass only if it interacts with the Higgs. In this sense, the Higgs "gives" mass to particles.