Letter cards, when performing a superior summon, gives your monster a new ability or changes the monster's ability when you summon it any with another monster that is, or hasn't been superior summoned already.
The new summoning method that uses these kinds of cards is called Superior Summons. It is the new kind of summoning method that links together the new letter cards, the monster you're using atm, and whatever summon method counterpart you use to superior summon in a duel.
As an example, when doing a superior summon with a ritual summoned monster already on the field ...or a Superior Ritual Summon, by using an invincible letter card you can bypass any ritual spell or field spell cards that are guaranteed to oppose your chances of superior ritual summoning a ritual-summoned monster in a regular duel. Better to have one invincible letter card than none at all if you're willing to make the move without a field or regular spell card.
A letter opening card can be used when doing superior synchro summons, but unlike the Invincible letter cards, letter opening cards serve a purpose related to synchro summoning rather than ritual, and that purpose is, it can only make your synchro summoning easier, and Plus, letter opening cards can't be used like the invincible letter cards, in other words you can't superior ritual summon a monster that's already in the process of becoming a superior synchro. There are ways to change superior summoned monsters abilities tho and that is by using a blank letter card, to either change your superior summoning or any summoning method you like or undoing your superior summon and turning it into a standard/normally summoned card.
letter closing cards can be used to finish, or complete a superior XYZ summon, like superior synchros superior XYZ summons lets you allow any monster to become your summoning material, no matter how high ranking in power level (attack or defense) the monster is.
Otherwise, when not using the letter cards as superior summoning materials the monster that's superior summoned using just a letter card all by itself will just become a normal superior summoned card, with only one extra special ability besides its current one, rather than possessing two extra special abilities when using the letter card counterparts.
And beware of chaos letter cards. They are used to summon chaos letter versions of the monster you are choosing to superior summon, in any way shape or form.
BEST O LUCK IN YOUR NEWFOUND SUPERIOR SUMMONS BROUGHT TO YOU BY YGO PRIME -L THE LATEST YGO SERIES BESIDES ARCV THAT FEATURES A FEMALE PROTAG (FINALLY) XD

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YUGIOH PRIME-L (THE FIRST DRAFT)
FanfictionCARDGAMES ON RAINBOW FARTING UNICORNS AND A FEMALE PROTAG WHO HAS NO CLUE ABOUT HOW TO DUEL OH AND THREE MYSTERIOUS MEN WITH MAGICAL EYE CANDY ITS TIME TO DUEL! IN D'ARTAGAR ;)