"soldering paste" : It is typically a name given to flux in a paste form.
However I've also seen people use it when they mean "solder paste" which is a completely different beast.
"Solder paste" is a suspension of small balls of solder in flux. When heated the flux activates, then the balls melt, and a joint is formed.
In this respect, solder paste is like rosin cored solder inside out, Instead of the solder containing the flux, this is the flux containing the solder.