What exactly is the Action? (Learning lagrangian) I have been trying to wrap my head around lagrangian mechanics but I find some parts confusing. For example, what exactly is action and why is it defined by the Kinetic energy minus the potential energy? This equation seems weird, wouldn't $E_k + E_p$ make more sense? 
Is it some sort of work done? I sort of get that it works and derives newtons equations(?), but I cannot connect it to the real world. 
So what exactly is action? Wikipedia does not do a good job of explaining or I just can't grasp it. 
 A: The only reason to define things as they are in classical mechanics is that they give rise to the correct equations of motions that can be directly measured and observed. Given a field theory described by $\phi(x)$, its action is defined as
$$
S(\phi,\dot{\phi})=\int_{\mathcal{D}}d^4x\,\mathcal{L}(\phi(x),\dot{\phi}(x),x)
$$
and the equations of motions for the fields are $\delta S=0$. This is an experimental result that can be observed at any level: classical mechanics, optics, electromagnetism, classical field theory, quantum mechanics, string theory and so on and so forth. In the case of a point particle in classical mechanics, the function 
$$
\int d^3x\,\mathcal{L}(\phi(x),\dot{\phi}(x)) = L(\phi(t),\dot{\phi}(t),t)
$$
(referred to as the Lagrangian) turns out to be $L=T-V$ because in this way it generates the correct Newton's equations of motion, no other reason than that.

This equation seems weird, wouldn't E_k + E_p make more sense?

Why would that make more sense? That is the total energy of the particle, which is a quite different thing (although it can be related).

Is it some sort of work done? I sort of get that it works and derives newtons equations(?), but I cannot connect it to the real world

There is plenty of literature deriving the Lagrange equations starting from the Newton's laws, passing through the principle of virtual works taking into account constraints and degrees of freedom and so on. Usually the course in analytical mechanics covers extensively such topics; however, the action principle does come from the real world indeed, making use of a little formalism and generalisation.
A: I'm not aware of a deeper explanation of action from a classical mechanics point of view. In the classical view, the action is just a functional of the trajectory which is minimized (or maximized) at the physical trajectory.
From a quantum mechanical point of view, and particularly the path integral formulation, there is a more physical interpretation: the action is essentially the accumulated quantum phase along a trajectory. Trajectories whose phase (action) is not an extremum (maximum or minimum) destructively interfere with nearby trajectories and so the only surviving trajectory is that with and extremum in the action.
From this point of view, Euler, Leibniz, et al presumably could have postulated quantum mechanics based on the principle of least action, but I'm sure no one would have believed them.
