What is an antenna amplifier ? How does it add gain?
An antenna has a gain set by its construction. For instance, a simple wire forming a dipole will have a gain of about 2dB whereas a dish might have a gain of 20dB. The lower the antenna gain the less directional will be the transmitted electromagnetic wave. Higher gain antennas such as dishes have to be pointed in the correct direction for them to be effective but, give you a bigger signal when receiving or concentrate the power output more when transmitting.

You can't affect antenna gain by adding an amplifier.


  if "for example" the Tx has a power of 20 dBm , antenna has a gain of
  10 dBi &amp; antenna amplifier "repeater" has a gain of 60 dB "like common
  cellular repeaters" , what does this gain mean ? is it mean the
  antenna gain will be 10+60=70 dBi ? or it mean the Tx power will be
  20+60=80 dBm ? I am confused !



The antenna gain is fixed by it's construction
Gain is a word that describes how much something amplifies a signal.
Tx power is 20dBm irrespective of antenna gain BUT this power is projected into a tight beam on higher gain antennas therefore it is equivalent to an isotropic antenna (transmits the power equally in all directions) fed from 20dBm increased by 10dB = 30dBm.



  another example for Rx antenna , if we have receiver has a sensitivity
  of -114 dBm and Rx antenna has a gain of 10 dBi &amp; we connect it with
  40 dB amplifier , should it mean the total sensetivity will be
  -114-10-40= -164 dBm ?"ignoring SNR please"


It's impossible to ignore SNR - if a receiver has an input sensitivity of -114dBm then pre-amplifying the signal by 40dB will likely increase noise and signal together and this will mean exactly the same SNR and no net benefit.


  if we use Tx has a power of -30 dBm at 1000 m, antenna has a gain of
  10 dBi &amp; at 1000 m distance there is a repeater has a gain of 60 dB ,
  how to calculate the total gain at this situation ?


You need to draw a sketch of something here - where is the final receiver (and I don't mean the repeater). Maybe something like this but with ALL the information on: -