A split air conditioner is will generally have an indoor component (evaporator) and an outdoor component (condenser).
When the compressor runs, the evaporator will get cold and the condenser will get hot.
A heat pump also has a indoor and outdoor components, along with a new item called a reversing valve. The reversing valve allows the system to switch which component is the evaporator and which is the condenser.
When the compressor runs and the reversing valve is enabled, the evaporator is inside and you get cold air.
When the compressor runs and the reversing valve is disabled, the condenser is inside and you get hot air.
T156 is right, most of the time there is just a catch pan and a drain pipe that dump the water somewhere else.
For more detail, condensers and evaporators get their names from what is happening to the refrigerant in the system. In the summer, the evaporator (inside) will get cold and cause water condensate to form, coming from the air.
This video by Technology Connections is one of the things being referenced in this thread. I haven't seen this one specifically, but he is very good at breaking down topics into pieces that slowly build on each other.