If you have some background in probability, the following family of examples may be useful. Let $X$ be a random variable with a continuous distribution, and probability density function $f(x)$. Let $\mathcal{A}$ be the plane region below $y=f(x)$ and above the $x$-axis. Recall that $\mathcal{A}$ has area ("weight") $1$.
A midpoint of $\mathcal{A}$ is a point $m$ (commonly uniquely determined) such that $\int_{-\infty}^m f(x)\,dx=\int_m^\infty f(x)\,dx=\frac{1}{2}.$ Such a point $m$ is called a median of $X$.
By way of contrast, the centroid of $\mathcal{A}$ is the point $\mu$ (if such a point exists) with the property that if we think of the whole weight of $mathcal{A}$ as concentrated at $\mu$, then this has the same moment about the $y$-axis as the actual region $\mathcal{A}$. It turns out that $\mu=\int_{-\infty}^\infty xf(x)\,dx,$ so the centroid of $\mathcal{A}$ is the mean of $X$.
For symmetric regions, the concepts more or less coincide, apart from some technical issues. But in general, the midpoint of the area is not equal to the centroid. For example, consider the exponential distribution with parameter $\lambda$. So $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$, and $f(x)=0$ elsewhere. Then $m=\frac{\ln 2}{\lambda}\qquad\text{and}\qquad \mu=\frac{1}{\lambda}.$ So in this case $m<\mu$. Informally, $\mu$ is more sensitive to area in the tail than $m$ is.
There are some interesting extreme examples. For instance, let $X$ have density function $f(x)=\frac{2}{\pi}\frac{1}{1+x^2}$ (for $x \ge 0$). Then the midpoint $m$ of the area turns out to be $1$, while the mean $\mu$ does not exist, or more informally is infinite. The "tail" goes down so slowly that though the area is $1$, the centroid is "at infinity."