This refers to the amount of money, in cash or securities, a financial institution is holding on behalf of its clients. The entity is a custodian, allowed to buy and sell assets based upon contractual terms (ToS).
Investment firms are obviously looking to increase their assets under management. Often, compensation is on a percentage basis, thus increasing as the AUM goes up.
The figure is derived by adding up all the assets held in client accounts.
For many, AUM is a metric that determines the viability and success of an institution. Many of the largest financial firms are ranked based upon their AUM. This is common in the institutional banking industry. Wall Street puts a great deal of emphasis on this.
Companies that manage money on behalf of their clients and run the portfolios are institutional investors.
Related
See List of Top 20 Asset Management Firms ranked by assets under management