The concavity part: The curve is concave up (facing up) in the region where $f''(x)\gt 0$, that is, where $x\gt 3$. The curve is concave down in the region where $f''(x)\lt 0$, that is, for $x\lt 3$. Concavity changes at $x=3$, so at $x=3$ we have a point of inflection.
Critical points: The critical points of $f(x)$ are the points where the first derivative of $f(x)$ is $0$ (or undefined).
We have $f'(x)=3x^2-18x+24=3(x^2-6x+8)=3(x-2)(x-4)$. Thus $f'(x)=0$ at $x=2$ and at $x=4$. These are the critical points.
Critical points are useful for identifying local maxima and minima of our function.
It is not too hard to see that $3(x-2)(x-4)\gt 0$ if $x\lt 2$, and also if $x\gt 4$. And $f'(x)\lt 0$ if $2\lt x\lt 4$.
So our function $f(x)$ is increasing in the interval $(-\infty,2)$. It is decreasing in the interval $(2,4)$, and increasing in the interval $(4,\infty)$. We reach a local maximum at $x=2$, because the function has been increasing until then, and starts to decrease. We reach a local minimum at $x=4$.