In plain English:

• The Beta distribution is a family of continuous probability distributions.
• It describes random variables that can take values anywhere between 0 and 1.
• One example of a beta distribution is the uniform distribution on [0, 1].
• Beta distributions are proportional to $x^{a-1}(1-x)^{b-1}$ where $a$ and $b$ are parameters. Setting $a=b=1$ yields a uniform, since the density is constant.
• Like all PDF's, these ones must have a total probability of 1. To get this to work, you have to divide $x^{a-1}(1-x)^{b-1}$ by its own integral between 0 and 1. This integral, seen as a function of $a, b$, has a name because it has other uses in mathematics. It is called the Beta Function.

In plain English:

• The Beta distribution family is a set of continuous probability distributions.
• It describes random variables that can take values anywhere between 0 and 1.
• One example of a beta distribution is the uniform distribution on [0, 1].
• A beta distribution has density proportional to $x^{a-1}(1-x)^{b-1}$ where $a$ and $b$ are parameters. Setting $a=b=1$ yields a uniform, since the density is constant.
• Like all PDF's, these ones must have a total probability of 1. To get this to work, you have to divide $x^{a-1}(1-x)^{b-1}$ by its own integral between 0 and 1. This integral, seen as a function of $a, b$, has a name because it has other uses in mathematics. It is called the Beta Function.