What's mean these microcontroller pin input/output modes?
Floating: neither pull-up nor pull-down. Your input goes effectively straight to the gate of a FET somewhere. Should be used with external driver or pullup/down only; don't leave it entirely floating to pick up ESD.

Analog (output): not quite clear from the datasheet, but I'd assume this was the output of a DAC somewhere and can therefore take a range of voltage values.

Open-drain: if set to "0", a FET will be activated to connect this to ground. If set to "1", it will be left floating. Used for I2C and similar buses where there are multiple open-drain drivers and a single pullup resistor somewhere.

Push-pull: contains both high and low output transistors. Fast, capable of actually driving things which require a positive voltage, but must not be connected to other drivers. If set to "1", connects the positive supply rail to the output via a FET.