How can the universe be hot or dense in the first moments after the big bang when it has no matter? From the CERN website:

In the first moments after the Big Bang, the universe was extremely hot and dense.

I've always heard this about the big bang but I've never thought about it before now. If "heat" is vibration of atoms and "density" is the amount of matter in a unit of volume, what do these terms mean in the context of a universe that doesn't yet have any matter?
 A: Fundamentally, "matter" is manifestation of energy excitations of a universal, all-encompasing substrate (the vacuum or "aether" or spacetime - properly understood, these all refer to the same unified framework of modern physics). The matter manifested as massive particles - electrons, baryons or their discrete combinations (nuclei and atoms) - - is "quantized", ie. you need at least E=mc^2 of energy to create one quantum of particular type (ie., an elementary particle) characterized by its rest mass m.
When the temperature in energy units kT is much less than the energy mc^2 of a single quantum, the number of quanta (e.g. atoms) remains constant and only their kinetic energy can redistribute. This is the low-energy (conventional) physics limit that the OP refers to. In the opposite limit of kT >> mc^2 all types of matter (with correspondingly small elementary masses) behave as radiation ("kinetic" energy dominates over the rest energy). This is the so-called ultra-relativistic limit, relevant for all matter in the early enough stage of the Big Bang.
"Density" is just the energy density. In the non-relativistic limit (e.g., for atoms in the air), we can keep track of "mass density" separately from the kinetic energy because the identity of atoms is consevred. In the ultrarelativistic limit, there is no difference between "dense" and "hot" - if it is hot enough, density and temperature are equivalent.
A: Remember that when we talk about the big bang, all we can actually do is take the current universe and extrapolate the clock backwards in time. We do this using General Relativity ($\Lambda \mathrm{CDM}$ for instance), and this leads us to a universe that is a uniform soup of extremely dense plasma in all directions, but after that that theory breaks down and cannot be used validly anymore. If we use a "dotted line" so to speak to extrapolate the existing theory back one more fraction of a second, we get a singularity where everything is compressed to zero size. But that is not physical.  It's kind of like saying the Ideal Gas Law predicts that at 0 Kelvin, the gas has zero volume.  What actually happens in real life is, the gas condenses into another state (liquid, solid, Bose-Einstein Condensate), and a completely different physical theory takes over. We haven't found this theory for the universe yet. But there is no reason to believe there was ever a point with "no matter or energy." Some hypotheses believe that there was a time before the hot plasma soup where everything was contracting from a more spread out state.
A: In Physics, hot is an adjective meaning at high temperature. Heat is a different concept. In the case of ordinary matter, the temperature can be associated with atomic speeds. However, it is possible to generalize the concept of temperature to deal with systems different from moving particles.
The most general thermodynamical definition of temperature is based on the rate of change with the energy of the number of the microscopic states of a system. Quite an abstract definition but sufficiently general to allow using the word temperature in cases where the classical concept of atomic movement cannot be used (quantum systems, or electromagnetic fields, to cite a couple of examples).
In the context of the Big Bang, the concept of density of the universe also needs a generalization of the usual idea of the mass per unit volume. The proper context for Big Bang theory is General Relativity (GR). Relativity tells us that mass and energy are not separate concepts: an increase in energy implies an increase in mass. Therefore, extremely dense is equivalent to saying that a huge amount of energy was confined in a very small volume.
Unfortunately, in many cases, the popularization of Science uses common language words without explaining that they may have a different meaning in Physics.
A: The term "Big Bang" had multiple usages. Two usages that are most commonly used are the following.

*

*The Big Bang is the state of the universe at a time when earlier time is much influenced by quantum physics. This usage is based  on the fact that at the present time science has not yet found a way to combine General Relativity and quantum physics.


*A somewhat less common usage is that the Big Bang occurs at the beginning of time, i.e. t = 0.This usage implies that we cannot know anything about the Big Bang state because it is a mathematical singularity which has no meaning in cosmology.
