State variable is used to refer to the state of the system at any arbitrary time; for example:

in the state(x dot=Ax+Bu)-space(y=Cx+Du), system; x is called the state, y is the output and u is input (control law), A,B,C,D are n*n, n*p, q*n, and q*p constant matrices accordingly.

Controllability is one where o/p can be controllable with change in i/p.

where as observability is simply observing what is happening inside the s/y with change in input.

The state of a system is the set of quantities that, once determined at a point in time, the future output of the system is completely independent of the past inputs of the system.  

state variable of the system are those parameters of interest  which can be used to find all other parameters of the system and whose knowledge allow us to know about the current or future state of the overall system.....

It is a good question especially for students. State space conjures up some difficultyfor first timers. IMHO, intuitively state refers to all the attributes needed to describe an entity.  As an analogy e.g. we may consider the "state of the country" address by heads of countries identifying economy. health, education as variables. At a more basic level for a student her  state wrt exam  could be described by variables such as topics known ,ability to solve problems, assignment completed etc.. In the case of a control system this translates e.g a moving vehicle at a point in time to 1. x1=location    2 .x2= velocity   3. x3=.acceleration   4. x4=shock etc .Notice in this list that x2 tells how fast x1 is changing and x3 how fast x2 is changing  etc.  This can be described for systems using matrix notation and we obtain the state space set of equations. Notice also the vector[ x1, x2 ] etc in a linear system satisfies  vector space definition. Controllability refers to the fact that  the output can be controlled by manipulating certain of the state variables e.g in a  balanced bridge circuit , it is not possible to  control the current in the "horizontal" arm by changing the applied voltage

 Cheers

state variables is the set of variables that you use them to describe your system attitude in mathematics.

Let's give an example : suppose our system is an object moving in one-dimensional direction, and we want to find its position at any time. In the case where the object is at rest it is clear that its position is known at any time "initial position" hence the position is the only state needed to describe our system. However once the object starts moving, knowing the the position at present time is not sufficient to describe the motion of the object at future time, hence another state is required in this case it is the speed. Again if the speed is not constant , the two states "position and speed" are not sufficient to describe the motion of the object, therefore a third state is required "the acceleration". As a conclusion, the states of a dynamical system are the set of variables that are necessary and sufficient to fully describe the system and determine its future behavior. In terms of math, if a dynamical system is described using differential equations, all the variables together with their derivatives are considered as states.

A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system.

https://en.wikipedia.org/wiki/State_variable#:~:text=A%20state%20variable%20is%20one,external%20forces%20affecting%20the%20system.

Best regards

Check of these links:

https://www.tutorialspoint.com/control_systems/control_systems_state_space_model.htm

https://www.site.uottawa.ca/~rhabash/ELG4152L305.pdf