'The Dragon Boat Festival' is the English name given to 端午节 the Duanwu Festival, which falls on the fifth day of the fifth month of the Chinese calendar, and which has other Chinese names. People celebrate this festival for different reasons; the following is the main one.
Back when China was seven kingdoms, a poet named Qu Yuan 屈原 (c. 340 - 278 BC) served as a minister of the Kingdom of Chu 楚国. Qu Yuan pointed out that Chu ought to attack the neighbouring Kingdom of Qin 秦国, which was growing stronger by the day, but the King of Chu banished him. Chu was later destroyed by Qin.
Incidentally, the succeeding King of Qin and eventual First Emperor of China, Ying Zheng 秦始皇嬴政, then unified the other kingdoms and built the Great Wall of China by linking their city walls together. 'Qin' is the word from which 'China' is derived.
Qu Yuan was overwhelmed with grief when he heard of the destruction of Chu. He tied a large stone to himself, and flung it into the Miluo River 汨罗江.
Onlookers rowed themselves out in boats to search for him, but they couldn't find him, so to stop the fishes from eating him, they boiled rice with meat and packed this into bamboo before flinging it into the river.
To commemorate Qu Yuan's life, the ritual was repeated yearly and gradually, people started eating the rice and meat, and wrapping them in leaves to form zongzi 'dumplings' 粽子 of which there are many recipes, shapes and sizes. My grandmother made pyramidal, rice-only versions, while my aunt loves fish in hers. Here's what a dumpling looks like, with and without its leaf-wrap. The savoury pork- and chestnut-filled versions taste even better than they look :-).
As for the search for Qu Yuan, it morphed into Dragon-Boat Racing, which is a fun hobby in the UK (the photo is from Wikipedia (but having said that I had a whale of a time banging the drum :-)) and taken very seriously and at a large scale in the Far East.
I'm off to grab some dumplings now; Happy Dragon Boat Festival!