What does it mean to say that "the fundamental forces of nature were unified"? It is said that immediately after the Big Bang, the fundamental forces of nature were unified. It is also said that later they decoupled, becoming separate forces.
Indeed, if we look at the list of states of matter on Wikipedia we see:


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*Weakly symmetric matter: for up to $10^{−12}$ seconds after the Big Bang the strong, weak and electromagnetic forces were unified.


*Strongly symmetric matter: for up to $10^{−36}$ seconds after the Big Bang, the energy density of the universe was so high that the four forces of nature — strong, weak, electromagnetic, and gravitational — are thought to have been unified into one single force. As the universe expanded, the temperature and density dropped and the gravitational force separated, a process called symmetry breaking.

Not only is it said that the forces were once unified, but this is also somehow related to the states of matter.
I want to understand all of this better. What does it truly mean, from a more rigorous standpoint, to say that the forces were unified and later decoupled? How this relate to the states of matter anyway?
 A: When we say that the forces were unified, we mean that the interaction was described by a single gauge group. For example, in the original grand unified theory, this group was $SU(5)$, which spontaneously broke down to $SU(3) \times SU(2) \times U(1)$ as the universe cooled. These three components yield the strong, weak, and electromagnetic forces respectively.
I'll try to give a math-free explanation of what this means. To do so I'll have to do a decent amount of cheating.

First, consider the usual strong force. Roughly speaking, the "strong charge" of a quark is a set of three numbers, the red, green, and blue color charges. However, we don't consider the strong force three separate forces because these charges are related by the gauge group: a red quark can absorb a blue anti-red gauge boson and become blue. In the case of the strong force, we call those bosons gluons, and there are 8 of them.
At regular temperatures, the strong force is separate from the electromagnetic force, whose charge is a single number, the electric charge, and whose gauge boson is the photon. There is no gauge boson that converts between color charge and electric charge; the two forces are independent, rather than unified.
When we say all the forces were unified, we mean that all of the Standard Model forces were described by a common set of charges, which are intermixed by 24 gauge bosons. These gauge bosons are all identical in the same way that the 8 gluons are identical. In particular, you can't point at some subset of the 24 and say "these are the gluons", or "this one is the photon". They were all completely interchangeable.

As the universe cooled, spontaneous symmetry breaking occurred. To understand this, consider slowly cooling a lump of iron to below the Curie temperature. As this temperature is passed, the iron spontaneously magnetizes; since the magnetization picks out a specific direction, rotational symmetry is broken.
In the early universe, the same process occurred, though the magnetization field is replaced with an analogue of the Higgs field. This split apart the $SU(5)$ gauge group into the composite gauge group we have today. 
The process of spontaneous symmetry breaking is closely analogous to phase transitions, like the magnetization of iron or the freezing of water, which is why we talk about 'strongly/weakly unified' matter as separate states of matter. Like the iron, which state we are in is determined by the temperature of the universe. However, a exact theoretical description of this process requires thermal quantum field theory.
