Rail to rail amplifier analysis problem
if we had used only for example nmos transistor and the DC level of
  vin1 was 0.2V (with VDD 1.8V) maybe the nmos transistor was off so the
  signal couldn't analyzed further.Is this true?


Yes. The FET needs some voltage between Gate and Source to turn it on. Below this threshold voltage it will be completely off and cannot amplify the signal. For example if the threshold voltage is 0.5V then for the differential amplifier to work properly both inputs need to be at least 0.5V above the negative rail. This problem does not occur at the positive rail because the Gate is 0.5V above the Drain so the FETs can still amplify right up to (or even slightly beyond) the positive rail.

A circuit using PMOS FETs has the opposite effect - it can work down to the negative rail, but not up to the positive rail. So you can get full rail-to-rail inputs by using both NMOS and PMOS amplifiers. The only problem is how to combine their outputs with the correct polarity and phase. In your example this is done using current mirrors, which convert the pull up outputs of the PMOS FETs into pull down outputs which are wired in parallel with the appropriate NMOS pull down outputs.