Why do constellation names change (eg from Cancer to Cancri) when used as part of a star's name? Cancri is what to Cancer?  or Leonis Minoris to Leo Minor?   Arietis to Aries ?  or Lyncis to Lynx.  

Are they the Flamesteed names, the Latin names, Greek names?
What is their name?
These are the Latin word endings. 

Latin nouns change their endings according to how the word is being used. English doesn't have this grammar. But note how the word "I" changes to "my". In Latin you would call "my" the genitive form of "I" The good people at https://latin.stackexchange.com/ can give more explanation. 

For example, "Leo" is a third declension noun and its genitive "Of the lion" is Leonis. Third declension nouns have a genitive that ends in -is.  On the other hand "Cancer" is a second declension noun with Genitive "Cancri". A Latin student spends much of their first year learning all these word endings.

When you write "Alpha Cancri" you mean "the alpha star of the Crab", because "of the crab" is cancri.