I am trying to learn the basics of Bayesian decision and I came across the phrase "proper prior" but I don't really understand what it means. Does anyone know?
A prior distribution that integrates to 1 is a proper prior, by contrast with an improper prior which doesn't.
For example, consider estimation of the mean, $\mu$ in a normal distribution. the following two prior distributions:
$\qquad f(\mu) = N(\mu_0,\tau^2)\,,\: -\infty<\mu<\infty$
$\qquad f(\mu) \propto c\,,\qquad\qquad -\infty<\mu<\infty.$
The first is a proper density. The second is not - no choice of $c$ can yield a density that integrates to $1$. Nevertheless, both lead to proper posterior distributions.
See the following posts which throw additional light on the use of improper priors issue and some closely related issues: