With each Normal Memorial Coin or Special Memorial Coin used, your Memorial Level decides which past Gacha rare items you can win and how successful you will be able to win a rare item in Memorial Shop.
Memorial Level starts at Lv.0 and is based on your achievement points total. For every 300 achievement points on your total, your level goes up by 1, up to a maximum of Lv.24.
With each Memorial Level increase, the Normal Memorial Coin's and Special Memorial Coin's winning rare success rate increases. Memorial Level also determines the range of rare items that can be won from the pool in Memorial Shop. Lower Memorial Levels will only be able to win rares from a smaller pool while higher Memorial Levels will be able to win from a larger pool (at Lv.24, any rare item in the pool can be won). The pool is determined by a table, which can be viewed in the Rare prizes section.
Note that the success rate of Premium Memorial Coin is not affected by the Memorial Level. Premium Memorial Coin also has no restrictions for which past rares it can win from inside the pool of Memorial Shop (effectively has Lv.24 range).
Your Memorial Level and Achievement Points total is displayed on the top of the Memorial Shop menu.