Risk weighted assets is a measure of the amount of a banks assets, adjusted for risk. The nature of a bank's business means it is usual for almost all of a bank's assets will consist of loans to customers. Comparing the amount of capital a bank has with the amount of its assets gives a measure of how able the bank is to absorb losses. If its capital is 10% of its assets, then it can lose 10% of its assets without becoming insolvent.
By adjusting the amount of each loan for an estimate of how risky it is, we can transform this percentage into a rough measure of the financial stability of a bank. It is not a particularly accurate measure because of the difficulties involved in estimating these risks. These difficulties are exacerbated by the motivation banks have to distort it.
Risk weighting adjusts the value of a asset for risk, simply by multiplying it be a factor that reflects its risk. Low risk assets are multiplied by a low number, high risk assets by 100% (i.e. 1).
Suppose a bank has the following assets: £1bn in gilts, £2bn secured by mortgages, and £3bn of loans to businesses. The risk weightings used are 0% for gilts (a risk free asset), 50% for mortgages, and 100% for the corporate loans. The bank's risk weighted assets are 0 × £1bn + 50% × £2bn + 100% × £3bn = £4bn.
Basel I used a comparatively simple system of risk weighting that is used in the calculation above. Each class of asset was assigned a fixed risk weight. Basel II uses a different classification of assets with some types having weightings that depend on the borrower's credit rating or the bank's own risk models.
Banks have a motive to take on more risk. If they win their bets, the shareholders (and management) take the profit, if they lose then the loss us likely to be shared with debt holders or governments (as banks are rarely allowed to fail). Part of the motivation for Basel II was that banks were able to work around the Basel I system by selecting riskier business within each asset class. Given this it seems remiss to have allowed the banks to use their own risk models, especially given that model risk was quite high even without the incentives the banks had to manipulate the models to understate risk.