Can somebody teach me or guide me on how to download MODIS data (NDVI) that covers southwest Africa so I can use the data for Nigeria as a region of area of interest.
For the automated download of MODIS data, you may check pyModis which is a Free and Open Source Python based library to work with MODIS data (http://www.pymodis.org/). It greatly simplifies the download and processing of MODIS time series. In the near future pyModis will be updated again to run even faster.
Then you can use the downloaded and preprocessed MODIS data easily in "any" GIS.
First you need to select the MODIS product you need. For what you said, MOD13Q1 is a god choice [0]. If you are not sure, you can check this guide [1]
Next you need to select the study area. You can use the map in this webpage [2] to get the ID of the MODIS tiles you need. For example, h18v08.
Then, you need to download the data. I am a Linux user, so, I used WGET to do it. Basically WGET creates a local copy of the NASA website [5]. You can apply filters by file name to select specific time periods or tiles. You can use this shell script as an example [3]. Besides, this script downloads in parallel using other Linux tools [4].
Once you have the data, if you install the GDAL library[6] with HDF support enabled, you can read the data with many GIS or statistical software; for example, QGIS[7] or R[8]. NASA also provides viewer software as well [9] which do not require GDAL. I think any of these software allow you to extract the data you need.
You can use AppEEARS tool by USGS where you can input the shapefile of area of interest, you can reproject the data and specify the dates, years and product you want to download. Please see these links.