Fibre Based Inkjet Papers are made of a substrate, usually cotton rag fibres or high grade wood pulp, upon which an ink receptive coating is directly placed. This means the character of the underlying paper tends to show through the ink receptive layer, allowing the character of the paper to add to the expressive qualities of the print.
The ink receptive layer and paper substrate are in direct contact in these papers, which can typically accept a higher ink load, meaning stronger blacks and greater colour saturation can be achieved.
Baryta is a word you'll often hear with respect to fibre based papers. This was a layer used in traditional silver gelatin darkroom papers between the paper base and the light receptive coating. These papers had a unique look and feel that many regard as the pinnacle of photographic expression.
Recent inkjet papers have come very close to this appearance, such as Hahnemuhle Baryta FB, Ilford Gold Fibre Gloss and Canson Baryta Photographique.
Resin Coated Inkjet Papers are the papers most people regard as 'photographic' because they're the type that your typical mini-lab uses. These papers are made using a layer of (usually low grade) wood pulp, over which a layer of plastic is placed, usually polyethylene. This forms a barrier between the paper substrate and the ink receptive coating that is placed on top to the plastic resin layer.
This use of a plastic layer makes for a very commercial, ultra smooth appearance. It does tend to strip the final print of any of the paper's original character though. Sometimes this clinical, commercial look has benefits, but very often it can make for a lifeless print.