A property is called local if it relates only to things very close to the point of interest.
For example, the function \(f(x)\) has a local minimum at \(x_0\) if \(f(x_0)\le f(x)\) whenever \(x\) is very close to \(x_0\). But there could be values of \(x\) with \(f(x)<f(x_0)\) when \(x\) is further away, as shown in this sketch.
Here, \(f(x)=x^3-3x\) has a local minimum at \(x=1\), as the point \((1,-2)\) is lower than all the points nearby. However, there are other points on the graph which are lower, such as \((-3,-18)\), so \(f(x)\) does not have a global minimum at \(x=1\).
A local maximum is defined similarly.