When your jeweller talks about these three aspects, to what exactly are they referring?
In a nutshell, they all relate to what you can see when light hits the facets of a diamond or gemstone. How much brilliance, sparkle and fire you see will depend on the cut of the facets. Facets make up the surface of the diamond or gemstone. When light hits and enters the facets it will bounce around just like it would do if it hit a series of mirrors. Where the light bounces depends on how well the cutter of the diamond sculpted the facets.
The better the facets can direct the light back up through the top for your eyes to see, the better the cut grade will be and the more brilliance and fire you will see.
Brilliance describes the total amount of light that is reflected by a diamond.
Fire describes the way the light is dispersed into different colours. The light bounces off the inside walls towards the centre of the diamond and then back up through the top of the diamond. The light is slightly bent by the diamond which creates a range of colours exiting the top of the diamond. The better the cut, the more fire you will see. 
Scintillation or sparkle works slightly differently. This refers to the way the light dances from one facet to another towards the centre of the diamond whilst the diamond is moving in the light. The diamond cutter has to direct the the light across the diamond as well us upwards in order to give it that sparkling effect.
This explains why the cut of a diamond or gemstone is so important.