Assets under management is the amount of clients' funds that a financial sector company such as a fund manager is responsible for managing.
This is a measure of the size of the business. It is often used to value fund managers, particularly when assessing takeovers, or the possibility of takeovers.
The value of a fund manager is also affected by other factors, most importantly the type of funds as this affects the level of fees. Active funds charge far more than index trackers (charges in both cases are usually a percentage of the assets managed). Hedge funds have more complex fee structures and it is usual for them to charge a (high) percentage of any gains.
Changes in assets under management can reflect both inflows and outflows of funds (i.e. clients investing or withdrawing investments) and gains and losses in the value of the assets. In order to understand underlying trends it is important to break down the changes in order to see whether the changes are due to gains or loss of clients funds (fund inflows and outflows) or investment gains and losses.
Investment gains or losses are, in themselves, not much of an indication of an underlying trend. The loss of client business probably is. Poor fund performance can do reputational damage, and thus affect the flow of business, potentially severely.