An improper prior is essentially a prior probability distribution that's infinitesimal over an infinite range, in order to add to one. For example, the uniform prior over all real numbers is an improper prior, as there would be an infinitesimal probability of getting a result in any finite range. It's common to use improper priors for when you have no prior information.

The mark of a good prior is that it gives a high probability to the correct answer. If I bet 1,000,000 to one that a coin will land on heads, and it lands on tails, it could be a coincidence, but I probably had a bad prior. A good prior is one that results in me not being very surprised.

With a proper prior, probability is conserved, and more probability mass in one place means less in another. If I'm less surprised when a coin lands on tails, I'm more surprised when it lands on heads. This isn't true with an improper prior. If I wanted to predict the value of a random real number, and used a normal distribution with a mean of zero and a standard deviation of one, I'd be pretty darn surprised if it doesn't end up being pretty close to zero, but I'd be infinitely surprised if I used a uniform distribution. No matter what the number is, it will be more surprising with the improper prior. Essentially, a proper prior is better in every way. (You could find exceptions for this, such as averaging a proper and improper prior to get an improper prior that still has finite probabilities and they just add up to 1/2, or by using a proper prior that has zero in some places, but you can always make a proper prior that's better in every way to a given improper prior).

Dutch books also seems to be a popular way of showing what works and what doesn't, so here's a simple Dutch argument against improper priors: I have two real numbers: x and y. Suppose they have a uniform distribution. I offer you a bet at 1:2 odds that x has a higher magnitude. They're equally likely to be higher, so you take it. I then show you the value of x. I offer you a new bet at 100:1 odds that y has a higher magnitude. You know y almost definitely has a higher magnitude than that, so you take it again. No matter what happens, I win.

You could try to get out of it by using a different prior, but I can just perform a transformation on it to get what I want. For example, if you choose a logarithmic prior for the magnitude, I can just take the magnitude of the log of the magnitude, and have a uniform distribution.

There are certainly uses for an improper prior. You can use it if the evidence is so great compared to the difference between it and the correct value that it isn't worth worrying about. You can also use it if you're not sure what another person's prior is, and you want to give a result that is at least as high as they'd get no matter how much there prior is spread out. That said, an improper prior is never actually correct, even in things that you have literally no evidence for.