Why can't I transform faces, edges, or vertices without transforming the whole object? My problem is that I sometimes lose the ability to independently translate or otherwise transform any parts of a single mesh. If I try, for example, to just translate several faces, edges, or vertices, then the entire mesh will stretch and displace itself in order to follow the translation. Or, alternatively, if I try to translate just one face then the whole mesh will move as if I'm translating in Object Mode.

I'm not really sure I know how to describe this problem properly. I don't even know how to reliably reproduce it. It seems to happen to any given mesh at random, and it doesn't stop happening to that mesh unless I delete and remake it.

I assume that I must be doing something wrong to cause this to happen. Does anyone have any ideas? Thanks.
You need to turn off proportional editing

Or use the shortcut: O key.
Proportional editing in action: