What are some examples of states of matter?

The so called classic states would be,

1) Solids - Definite shape and Volume. Molecules tightly packed with one another. Very strong intermolecular forces. Molecules can only vibrate about their mean positions.

2) Liquids - A definite volume but no definite shape. Assumes the shape of the container. Intermolecular forces intermediate between those of solids and gases. Molecules can move but the range of motion is limited.

3) Gases - No definite volume or shape. Extremely weak intermolecular forces. Molecules can run away from one another.

However, there are other phases as well. The most common would be PLASMA - A soup of ions excited to fantastic temperatures.

Other exotic phases exist. One of them worth mentioning is the celebrated Bose-Einstein Condensate which is a dense collection of Bosons cooled down to fantastically low temperatures very close to the absolute zero.
Such phases very first observed in 1995. A Rubidium gas cooled to 170 nK condenses to form such a phase where the individual identity of the atoms are lost. They behave like a dense collection or more appropriately a giant atom with all atoms collapsing to the lowest energy state.

Bose-Einstein statistics imposes no priori restriction on number of particles that can occupy a particular state (Pauli's exclusion principle doesn't apply to Bosons) and that's what makes it possible for a large number of them to condense into the lowest energy state when cooled to very low temperatures.