1- What's difference between General purpose output and Alternate Function output(in other word, when should i use of GPIO and when should i use of AF output)?
Alternate function: if you want to use any of the other peripherals connected to that pin, for example USART. I.e. use use alternate function if you don't want to use it as GPIO, but for something else.
2-
A) what's mean the Input Floating and the Input Pull-Down and the Output Push-Pull and the Output Open-Drain?
B) What's the usage of these and when should i use of these?
Input pull up/down: enable internal pull up or pull down resistors, so the port gets pulled up/down if nothing is connected to it. One example would be a push button connected to ground: you connect the button from the input to ground and enable the internal pull-up. If you press the button the input is pulled low through the button, if you release the button then the pull-up pulls it high. Input floating disables both the internal pull up and down resistors. Useful if the input if the other side is push-pull, since the pull up/down resistors would just be wasting power.
Push pull means that the pin is either driven high or driven low (the driver sources or sinks current), open drain means that it will pull down the output (sink current) if set to low, but will tri-state the bus (become high impedance, no current will flow in/out) when set high. This is used in I2C, for example, where multiple drivers can pull the bus low, and the bus is pulled high through the pull-up resistor if all devices are high. This prevents multiple drivers from fighting each other.
This is all very similar to how the AVR works, by the way, except that the AVR lacks pull-down resistors.
3- As you know, almost all of the ARM based MCUs haven't any eeprom for emergency situation. should i use of Backup register?
As you can find in the datasheet, the contents of the backup registers are preserved as the STM32 goes into a low power standby mode, they will not survive if it completely loses power.
The answers to all of your questions are in
this 1000 page document. Quite a bit larger than your average AVR datasheet.
This introductory book may also be useful. In general terms like push-pull, open drain, and pull up/down won't be discussed in documents about the STM32 since these are general electronics terms. A general digital electronics resource or Wikipedia may be helpful for those.