What is force? The basic intuition Conventional definition of force is :

An agency (push or pull) that tends to change state of rest to motion or vice versa. 

I don't find this definition satisfactory. 
Second definition comes from newton's second law :
$$ F = ma$$
But as feynman has said that this law relates two fundamental virtues and is used to define one in terms of other, which is not idea of physics ( I am not quoting the exact wording) 
Then what actually force is? I want to know basic intuition idea behind it (even in modern physics) 
 A: Force is defined as instantaneous rate of change of momentum, but it is not merely a product of mass and acceleration as said by Feynman also in his lecture :

The real content of Newton’s laws is this: that the force is supposed to have some independent properties, in addition to the law $F=m\mathbf a$ but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law$F=m\mathbf a$ is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.

It says that forces have material origin and not a vague concept,as it is seen evidently, like we often say that force is a pull or push, which implies that we have a physical notion attached to the concept and it greatly simplifies our quest to understand and predict nature.
Hope I made myself clear.
Thanks.  
A: One way to understand force is to say that the net force is defined as the rate of change - or time derivative - of momentum. That is exactly what a force is in all physics because that is what it is defined to be. Then Newton's second law is a definition. ($\bf{F} = m\bf{a}$ because the time derivative of momentum ($\dot{\bf{p}}$) is mathematically the same as $m\bf{a}$ for a constant mass.)
Remember that you can have forces which do nothing - because some other force counteracts them; Newton's Law talks about "net force." So a force is anything that contributes to the net force which Newton defined.
Now your question becomes "What is momentum?" Well, that changes a little in some fields of physics, but the typical definition is $\bf{p} = m \bf{\dot{x}}$ where $\bf{x}$ is position. That is, momentum is the mass times the time derivative of position.
Regarding, "an agency (push or pull) that tends to change state of rest to motion or vice versa" - that statement is intended to give you intuition. You should not find that definition satisfactory because it is not a good definition.
