This post explains the Vasicek/Merton single factor model which is part of the Basel framework (IRB approach) and has been used to evaluate CDOs.

Imagine you loan money to a friend who will default with a probability of 1%. When it comes to paying back the loan you will either receive 100% (plus any interest) or 0%. That's pretty risky. So you figure you can improve your situation by making same size loans to n friends. (NB: You will need a license for doing so). In case they all default individually with a probability of p = 1%, the law of large numbers tells you that the more loans you make (increase n), the closer the average default rate (= portfolio default rate) will be to 1%. "The" central limit theorem states that the portfolio default rate will be normally distributed with a mean of 1% and a variance that goes to zero as n increases.

Now here is the result for n = 5000 loans and 100,000 simulations (portfolios):

Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend. But in case you make loans to colleagues from work this assumption won't be correct since bankruptcy of the company would turn most of your loans sour at the same time no matter how large (number of obligors) your loan portfolio actually is. In other words: Certain systemic risk can't be diversified away.

Assuming you really have lots of "similar" friends (p = 1%) and they are all pretty evenly distributed across the sectors the economy has to offer one could argue that defaults are actually only a function of this idiosyncratic risk (as before) and a single systemic risk factor reflecting the overall state of the economy. In case the economy does very well, hardly anybody will default (even an expensive divorce is not an issue) and in case the economy enters into a deep recession, the default rate goes north. The sensitivity of each obligor to this systemic factor and the correlation among the obligors is given by √ρ and ρ, respectively.

Result for n = 5000 loans and 100,000 simulations (portfolios):

The average portfolio default rate is not affected by an increase in correlation, but the higher the correlation the more likely extreme portfolio default rates become (good or bad). In case the correlation is one, we are back to a single obligor. Either nobody (0%) or everybody (100%) defaults.

The model (see comment section for details) is a useful starting point but since both systemic and idiosyncratic risk is assumed to be normally distributed and are connected via an uncertain correlation coefficient you could easily be on the wrong end of the trade. NB: That doesn't mean the Basel guys did a bad job. They had two parameters (correlation and systemic shock size) for calibrating the model.