Direct Sampling from posterior distribution Why is direct sampling from the posterior distribution intractable?
 A: First, how do you sample from any probability distribution? The only way I can think of is if you have a random number generator from the target distribution. Then just generate these numbers and you have a sample. 
There are several methods to do this, but let's stick with the inverse method just as an example. As the wikipedia article notes, for continuous variables (distributions), we need to integrate the pdf, and this may be intractable for most distributions (integrals are hard to solve). 
So, we can't sample from distributions that we don't have a random number generator (rng) from those distributions. And in most Bayesian applications, posterior distributions are not from standard distributions, i.e., distributions that we have a rng. So, we can't sample directly from it.
A: In order to have a tractable posterior distribution, one has to set a conjugate prior distribution. 
It depends on the sampling distribution of the model. If it belongs in the exponential family of distributions, then you can choose a conjugate prior. The posterior that yields is in the same family of distributions with the prior distribution, and is tractable.
If the prior is non-conjugate then even if you can make direct calculations you will find a posterior that is of non-standard form.
