a) Informally, we may differentiate different states of matter by its appearance, mobility, etc. The descriptions are qualitative and sometimes confusing.

b) Using the classical scientific language, we can classify the states of matter into solid, liquid or gas by its symmetry. Those contrasting states of matter are the result of different symmetries. For example, the translation symmetry is preserved while the rotation symmetry is broken in solid. In contrast, the translation symmetry is broken while the rotation symmetry is untouched in gas. The liquid state lies in between these two.

c) There are new states of matter that are beyond the scope of classification by symmetry alone. We have to introduce some other terms such as the topology, etc. A good example is the topological insulator, which is surface conducting while the bulk is insulating. The topological insulator is clearly different from the regular insulator, but can't be differentiated by the language of symmetry only.

The four states of matter that we commonly deal with are solid, liquid, gas, and plasma. You personally have experience with three of the four of these: the ground that you stand on is solid, the water you drink is liquid, and the air that you breathe is gas. Plasma is different from the others in that it is affected by magnetic fields, which gasses are not, but otherwise behaves much like a gas. The sun is made out of plasma. What I don't understand is what you are referring to as "evidence" - you know how the solid ground, liquid water, and gaseous air behave around you, and how they are different. We can do experiments on them to know what their properties are in more detail, but the experiences you have with them are evidence themselves.