If you take something lovely like coconut oil, it's not so soluble in water. But if we sulfate it, it becomes more soluble. Combine that with something like sodium or ammonium - again, something very soluble - and we have something that works in water.
Sulfation is a bit like hydrogenation, when a molecule is manipulated by adding something to it. In the case of hydrogenation, extra hydrogen molecules are adding to an oil to make it less susceptible to rancidity. In the case of sulfation, a sulfur molecule is added to something to make it more soluble in water.
As a note, normally the sodium, potassium, or ammonium ion is positive - so that's the hydrophilic end - and the sulfate ion is negative - so that's the lipophilic end.
Point of interest: I'm going insane typing "sulfate". In Canada, we'd write "sulphate", so you'll see me go between the two. But most INCI listings show the word with an "f".
Let's take a look at the first of the sulfates - alkyl sulfates!