Foundation piecing is when sew your pieces onto a another fabric, like muslin. The string quilts that I make are on a foundation of muslin.
Paper piecing is when you have a pattern printed on paper. You sew your block pieces onto this pattern following the lines on the paper. When finished you remove the paper. It is good for intricate patterns and awkward shapes that you would have to cut and try to piece.
If you type these terms in search you will get lots of info, but this is the basic use of each.
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