Can somebody teach me or guide me on how to download MODIS data (NDVI) that covers Mongolia so I can use the data for Mongolia as a region of area of interest.
Zagarjav Tsegmed please follow the steps from Mohsen Ahmadkhani and download the MODIS tiles of spatial resolution that you are looking for. Once you have the NDVI tiles downloaded, you can use any GIS software to clip your study area from it (For instance you can use Clip raster by Mask Layer tool available in QGIS). You can use shapefile of Mongolia as a mask to clip the downloaded NDVI tiles. I hope this helps.
you can used Landsat 8 images from USGS, go to determined the study area by coordinate system in the area, then calculate NDVI by easy steps in ArcGIS, you available these steps by search in YouTube.
Modis has provided two good products of NDVI and EVI called MOD13q1 and MYD13q1 (Terra and Aqua platforms) with spatial resolution of 250 meters and temporal resolution of 16 days.
I think the two mentioned products are also available for your area of interest.