What is meant by the term 'state' in a finite-state machine?
A state is simply an assignment of all possible variables of a system to values. A state transition then means a transition from one possible set of values of variables to another set of values. 

To see how this might be useful, imagine a set of two switches $S$ and a light bulb $B$ that are connected to emulate some unknown logical gate. If you want to find out what that gate is, you can use a state machine model: 

Say $I = \{S,B\} $

Initial State: $I = \{\{0,0\},0\}$

State transitions: 

$\{\{0,0\},0\} \rightarrow \{\{1,0\},1\} \rightarrow \{\{0,1\},1\} \rightarrow \{\{1,1\},0\} $

It is evident that every state is just a set of different possible values of $I.$ 

By looking at the values of $B$ for a set of values of $S$, You can easily deduce that the system emulates a XOR gate.