The memory of an object is its ability to revert back to the shape that it was in. This could mean that if your line has been on a spool then it could come off the spool in the same shaped loop as it was on the spool. This is not ideal as you want the line to do whatever you want it to do and you want it to revert back to the straight shape that it was made as.
If your line has developed a curve then this can be partially solved by stretching out the line in order to try and get rid of the curve. If the line has started to deform then it might be the best time to throw it away and purchase a new line. The top brands prevent this from happening to ensure that you are left with a perfectly usable fishing line time and time again.