In your example, R1 is a pull-down and R2 is a pull-up resistor. Depending on the MCU and the pin involved there may be one or the other or both or neither available. That information will be in the datasheet. There's also another possibility, a "hold" where there is a resistor internally from a buffer output back to the input.
The purpose of a pull-up or pull-down is to put the input line in a known state if the connection to it is high-impedance. On an MCU that can happen if the wire gets disconnected or if the driver is deliberately tristated or during startup before it is configured. If the line is being driven push-pull it does little but waste power.
Whether a pull-up or pull-down is required is dependent on your requirements. As to whether the internal resistor is sufficient, again that depends on the requirements. The IC makers tend to choose rather high values which may not be desirable in certain circumstances where EMI or leakage is present. There might be cases where the values are too low (very low power systems, for example). The on-chip resistors (or equivalent) also have quite a loose tolerance typically. So there are many cases where a pull-up or pull-down is available on the chip, but the designer chooses to use an external resistor.