Game of Thrones: How Olenna Killed Joffrey

Joffrey was poisoned during his own wedding to Margaery. He asked Tyrion to serve him a drink. The drink turned out to be poisoned. 

Tyrion didn't knew the drink he was serving to Joffrey was already poisoned. 

Baelish wanted to kill Joffrey for his own reasons while Lady Olena didn't want her daughter Margaery to marry such a monster , the word Olena used to describe Joffrey. 

Dontos , the man Sansa spared his life before, Gave her a necklace as an act of gratitude. The Necklace was given to Dontos by Baelish and it had the poison in one of it's crystals in a solid form.

Dontos was following Baelish's orders and he knew he is being a part of a plot. This is why Baelish killed him after the mission was done. (See Game of Thrones: Why Cersei hates Tyrion?)

During the wedding Lady Olenna took the poison from Sansa's necklace , without the later knowing, then dropped it into Joffrey's wine. 

Sansa wasn't a part of the plot nor she knew what was going on. She really believed that the Neckalce was a gift from Dontos because she saved his life. (See Game of Thrones Season 3 Summary)