The simplest and most widely agreed upon definition of aging is also the least helpful: aging is a rise in the risk of death due to intrinsic causes, meaning the failure of one or more organs or systems vital to life. A person is more aged if his or her risk of death has grown over time.
For a better explanation than this we have to ask why and how aging happens. Broadly speaking there are two schools of thought on this matter. The first, dominant school argues that aging is caused by damage, forms of wear and tear to cells and tissues, and that damage leads to characteristic changes and failures in our biology. The second, lesser school argues that aging is an evolved program of characteristic changes and failures in our biology that in turn cause damage and eventual death.
It is a mark of how complex aging is under the hood that the research community can accurately measure all sorts of damage, changes, and failures that accompany aging, but still have room to argue over whether damage causes change or change causes damage.