This is a good question, and also a good reminder to be careful with one’s language when writing about physics!
Most of the time, if you’re reading an article, what we mean by mass and rest mass is exactly the same. Rest mass is a slightly more precise term in its phrasing, meaning specifically the mass of that object when it is at rest, relative to the person measuring its mass. This is almost always how we measure mass. If you’re in a lab (or a kitchen), and you measure an object’s weight on a scale, that object is not moving at any speed. If it is moving at a speed, you should probably catch it, because it’s rolling off your scale.
If you’re on Earth, you take the weight — which is the force with which that object is pressing on your scale, within our Earth’s gravitational field – and can then convert it into a mass. This mass you’ve measured is the “rest mass”, since nothing’s moving in this scenario. Generally this number — the rest mass — is the most useful metric of how much material is assembled into whatever you’ve measured, so “rest mass” is often abbreviated into just regular old “mass”.
However, there is a way to make an object sort of behave as if it has a much larger mass than its rest mass, and given my pointing out how nothing’s moving in the definition of rest mass, you will rightly guess that it has something to do with motion. However, it’s not just any motion — an egg rolling off your kitchen scale isn’t any heavier than the egg that managed to stay balanced.
You’d have to accelerate that egg to a significant fraction of the speed of light before the egg started behaving as though it were heavier (and if you can do that without crushing the egg, you’ve won the world’s most difficult Egg Drop Challenge). It’s critical to note that the egg itself is not intrinsically heavier, and so we’re careful to keep “rest mass” separate from this relativistic speed effect. However, because the egg now has so much kinetic energy from moving so fast, and there is an equivalence between mass and energy (hello E=mc²), the energy of the object can masquerade as additional mass by adding to that object’s momentum.
The faster our relativistic egg is moving, the larger this energy-mass bonus gets, and the egg gains more and more momentum. The relevant feature of momentum here is that it resists changes to its speed. In particular, for our relativistic egg, the faster and faster it goes, the more momentum it has, so the harder it is to speed it up any more – you need to start expending truly ludicrous amounts of energy to speed it up even a little bit. This is one half of the reason it’s physically impossible to accelerate an object all the way to the speed of light — the closer you get, the closer to infinite energy you need to continue speeding it up.
So you can safely substitute “mass” anytime you see anyone mention a “rest mass,” but there are situations in which an object might behave as though it has more mass than its “rest mass” — these are, however, limited to objects traveling a significant fraction of the speed of light.
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