The Normalized Differential Vegetation Index is a very popular vegetation index that gives us useful information about the presence or absence of live green vegetation.
NDVI is calculated using a band with red spectral reflectance values, and another one with near-infrared reflectance values. In the sample dataset, you will find two image files named red.tif and nir.tif that can be used to compute NDVI. A project named ndvi.qgs is available, which contains these two layers and a landsat image corresponding to this same area. Open this project.

red.tif layer in the Red Band field and the nir.tif layer in the Near Infrared Band field. Click on Run to run the algorithm.
All vegetation indices that are computed by the algorithm are based on the relation between red and near-infrared reflectances. Leaf cells scatter solar radiation in the near-infrared reflectance and absorb radiation in the red reflectance, which can be used to predict the location of healthy green vegetation based on these two values.
NDVI is computed with the formula given in the following section.
As the formula of the NDVI is rather simple, you can calculate it without using a specific algorithm, just by going to the raster calculator. You can use the one integrated in the Processing Framework or the QGIS built-in on. You can see how you should fill the parameters in the QGIS Raster calculator to compute the NDVI, based on the two proposed sample layers in the following screenshot:

The vegetation indices algorithm requires the red and infrared values to be in two separate layers, each of them with a single band. However, it's common to have both of them in a multiband image. To be able to use these bands, you must separate them, extracting them into two separate files.
This can be done using the GDAL translate algorithm. The project contains a multiband image named landsat.tif with the red band in band number 3 and infrared band in band number 4:
The Translate algorithm uses the GDAL library underneath. You can also use this library as an independent tool from the console. At the lower part of the algorithm dialog, you will find a text field where you will see the equivalent console call to your current algorithm configuration.