[morzh]

In short (I a sure you've seen the video Rich provided), safety caps are X and Y.
X caps are placed across the Line/Neutral to filter differential noise.

Y caps are placed between the isolated parts of device, one being at a dangerous voltage referenced to Earth (either AC line or any voltage directly derived from it while being galvanically coupled to it, like rectifier, boost converter and such) and the other being the user touchable end, usually chassis GND.
This caps are to filter common mode noise.
Because they go to User end, they are made so that they are guaranteed to always fail open, thus avoiding electrocuting the poor user. They are typically not very high in capacitance.