What are state space equations?
Frequency-domain transfer functions are one way of representing the behavior of linear time-invariant systems.  'State space' is another.  

In the state space representation, the system is represented as collection of coupled linear first-order differential equations.  There are four matrices (usually called A, B, C, and D) that define these differential equations and show how system inputs affect the internal state, how outputs are derived from the internal state, and how the internal state evolves with time.

I recently gave a short intro talk on 'state space'.  Maybe the slides will be handy for you.