That is the Combustion Integrated Rack (CIR) with its MDCA (Multi-user Droplet Combustion Apparatus) insert. This paper has a lot of technical details on CIR.
The MDCA is capable of dispensing and igniting various liquid fuels with different mixtures of oxygen and inert gasses to study combustion in microgravity.
There's no convection due to atmospheric density, so the flames are spherical as shown below.
(source video)
That portion of video you linked to was actually taken during Expedition 44/45 in 2015, since it shows astronaut Kjell Lindgren handling the MDCA. In between the time the original video clip was recorded and when NASA posted this video, the MDCA had been replaced by the ACME (Advanced Combustion via Microgravity Experiments) insert. The inserts were swapped on October 11, 2017.
Shown below is ESA astronaut Paolo Nespoli wit the CIR optics bench translated out and rotated for maintenance. This was still with the MDCA insert which was later replaced by the ACME insert on the same Expedition 52/53.
(source video)
The ACME insert (shown below) allows for tests of gaseous fuels. It was one of my favorite experiments to watch during a slow shift on console. Here's a video of a particularly interesting test point.