Diversifiable risk is simply risk that is specific to a particular security or sector so its impact on a diversified portfolio is limited.
An example of a diversifiable risk is the risk that a particular company will lose market share. It will not have any impact on other companies in a diversified portfolio, so the only loss to an investor holding shares in the company will be the decline in that one share.
On the other hand a rise in interest rates will reduce the value of all shares and bonds. This means that it is not possible to diversify interest rate risk away.
Which risks are diversifiable depends on what universe of investments you are looking at. National level risks become diversifiable if you buy shares in other countries, some risks to equities can be diversified by buying bonds or commodities. Models such as CAPM define risk as diversiable or not in the context of a particular universe of securities: in the case of CAPM the market used to calculate β and the expected market return.
Of course, non-diversifiable risks can be controlled by hedging. It is also possible to choose securities that are less exposed to non-diversifiable risks: for example, a portfolio that is overweight on defensives is less vulnerable to an economic slowdown, but at the cost of lower expected returns.