Cherish balls(the red ones) are only obtainable by events & are the pokeballs that give-away pokemon are caught in. Dream balls (the last one) are like master balls that only work in the dream world.
EDIT- The yello one (before last) is a friend ball & it is used In the pal park in d/p/pt/SS/hg & always catch the pokemon in the pal park. The one under the moon ball is the gs ball (i think) & is unobtainable in the games, it is an anime ball used to catch Celebi that was to be delivered from proffessor elm to professor oak by ash.
Hope this helps :)