Foundation Piecing- a usually light weight fabric that has printed or hand drawn lines on it that you sew fabric onto to create a pattern block. The "foundation" fabric is left in and the blocks are sewn together to make a top.
Paper Piecing-A design is printed or drawn on paper. Fabric is put on top of the paper and you stitch through the paper and fabric to create the design. The blocks are sewn together and the paper is torn off when the top is assembled.
English Paper Piecing-This is where a paper shape is cut from card stock or similar paper to a specific shape, hexagons, diamonds etc. Fabric is placed over the shape and the fabric is hand tacked to hold the shape. The fabric shapes are then hand sewn together, catching just the fabric edge and the shapes are removed when the top is done.
HOWEVER
Some people call paper piecing, foundation piecing. Some people do paperless paper piecing where the paper is put on top and the paper is folded back as the fabric is pieced, some people use freezer paper for paper piecing. It is like the names of a certain quilt block, different regions call it one name and it gets named something else along the way.