Two roads up the same mountain
Newton gave us a superb machine: draw the free-body diagram, add up the vector forces, and apply $\mathbf{F}=m\mathbf{a}$. It works — but it makes you track every force, including the constraint forces (the normal force, the string tension) that do no useful work and are often unknown until you solve the problem.
Lagrange's reformulation throws the vectors away. You describe the system with a single scalar function — the Lagrangian L — and one universal recipe produces the equations of motion, in whatever coordinates suit the problem, with the constraint forces silently taken care of. This is the language in which all of modern theoretical physics is written.
Generalized coordinates and configuration space
A generalized coordinate is any set of variables q_1,\dots,q_n that fixes the configuration of the system. They need not be lengths: for a pendulum the natural coordinate is the angle \theta; for a bead on a wire it is arc length. Choosing coordinates that respect the geometry is half the battle.
Collect the coordinates into a single point \mathbf{q}. It lives not in ordinary space but in configuration space — an abstract space with one axis per coordinate. As the system evolves, that point traces a curve; finding the motion means finding that curve.
One point in configuration space encodes the whole system's arrangement.
Degrees of freedom and constraints
The number of independent coordinates is the number of degrees of freedom f. For N particles that would be $3N, but each [[holonomic-constraint|holonomic constraint]] — an equation like g(\mathbf{q},t)=0$ tying coordinates together — removes one.
Degrees of freedom = spatial coordinates minus the number of holonomic constraints c.
The big idea: nature is economical
Newton's method steps forward instant by instant. Lagrange's method judges the whole path at once. Of all conceivable trajectories between a fixed start and end, nature selects the one that makes a single quantity — the action S — stationary. This is the principle of least action.
The action is the time integral of the Lagrangian along a candidate path.
That is the entire content of the theory in one line. The next guide turns this principle into a concrete equation you can solve; the rest of the track shows how far it reaches — into conservation laws, field theory, and quantum mechanics.