probability

When we estimate the posterior density function, we have the following equation:

$p(x|data) = \frac{P(data|x)*p(x)}{p(data)}$

Let us think that our prior is a continuous distribution, say normal. We are trying to get probability of success in a trial that is Binomial, that is p(x|data). Now, What exactly this multiplication means?

$P(data|x)*p(x)$ ?

P(data|x) - is 1 value, say 0.10, or 0.5, etc. On the other hand p(x) is the value of the continuous probability density function (normal here). How can we multiply the value of pdf with P(data|x)?

P(data|x) is the concrete realization of the binomial trial with some given parameter x. According to Bayes rule p(x) should be the probability of the parameter, but p(x) is density! Without integration it has no sense. Could someone explain please?