The ADP1720 says that the supply voltage is from 4v to 28v in order to make it work properly.
My question is that if now I use 3.3v for the input voltage, what will happen? Is the output voltage is 3.3v or something else?
I presume you are using the 3.3V fixed output version of this part. When the supply voltage drops below a certain point, the LDO enters dropout (in this case 4V is specified as a safe margin although the actual point may be lower depending on load and device temperature). In dropout, the LDO cannot maintain the desired output voltage. Instead, the output will track the input voltage minus the dropout voltage.
I would suggest you take a look at figure 12 of the datasheet at http://www.analog.com/static/imported-files/data_sheets/ADP1720-EP.pdf to understand how the input and output voltages relate (although note this graph shows a 5V output part). Figure 11 shows how the dropout voltage varies with output current.
For this device the dropout voltage for a load of 50mA is around 275mV so I would expect an output of 3.025V. For normal operation to ensure a regulated 3.3V output, you should stay within the 4-28V input range.
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