AUM is the total market value of the money a fund, or a whole fund house, manages on a given day.
Assets under management is the market value of everything a scheme holds, less what it owes. For a fund house, it is the total across all its schemes. On this site it appears as fund size.
AUM changes for two reasons:
Fund houses publish each scheme's AUM every month. Size can matter in a few ways:
A larger AUM does not by itself mean higher or lower returns. The screener lets you filter funds by size, and compare shows it next to returns and costs.
Each fund page has its returns, risk, costs, managers and everything it holds.