Raffinating something is a technique used in metallurgy to remove impurities from liquid material. There are many different kinds of raffination, for example you can use vacuum to extract hydrogen from metals.
In solvent extraction, a raffinate is a liquid

Liquid

Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

 stream that remains left after the extraction with the immiscible liquid to remove solutes

Solubility

Solubility is the property of a solid, liquid, or gaseous chemical substance called solute to dissolve in a solid, liquid, or gaseous solvent to form a homogeneous solution of the solute in the solvent. The solubility of a substance fundamentally depends on the used solvent as well as on...

from the original liquid. From French raffiner, to refine.
In solvent extraction, the two liquids must be immiscible. This will allow separation of the product rich extract and the depleted raffinate.