What you are looking for is the amazing [`execute`](https://minecraft.wiki/w/Command#execute) command:

> Executes a command on behalf of one or more other entities, with originating permissions, optionally on condition that a single-block /testforblock-style check passes.

The syntax is:

    execute <entity> <x> <y> <z> <command …> 

`<entity>` is a target selector, such as `@a`, `<x> <y> <z>` are coordinates from which to execute the following `<command …>` from. In most cases, setting this to `~ ~ ~` (i.e. the exact location of the entity) is fine. To set the block under every player to air, you can use

    execute @a ~ ~ ~ setblock ~ ~-1 ~ air