You can have multiple stuns, slows, snares, etc all affecting you at the same time. They all apply their full effect for their full duration.
That's it, it's that simple.
So a stun does not replace a snare - you are both stunned and snared but since stun stops you doing anything including moving the snare is useless in this case.
For example lets say you are hit by a 1 second stun. 0.5 seconds later you are hit by a 2 second snare. 1 second after that you are hit by a 3 second slow.
In that case you are:
- Stun hits: Stunned only for 0.5 seconds
- Snare hits: Snared and stunned for 0.5 seconds
- Stun ends: Snared for 0.5 seconds
- Slow starts: Snared and slowed for 1.5 seconds
- Snare ends: Still slowed for 1.5 seconds
Even if it is multiple CC of the same type this happens the same way. If you get stunned for 2 seconds then after 1 second hit with another stun for 2 seconds, then you get 3 seconds stunned. This is what people mean when they talk about chaining CC. You hit the target with a new CC by the time the first CC has ended.
When combining slows you do not get the full strength of multiple slows at once. Additionally fears, charms and taunts all force you to move in a specific direction so again they may combine in strange ways. However the general rule here should be clear.