Lughnasadh (don't even try the nuances of the Gaelic pronunciation just stick with LOO-nus-uh) is a harvest feast and games held at the order of the Celtic god Lugh to honor his foster mother Tailtiu. Tailtiu may have been a mother earth goddess. She died at the harvest and told her followers to remember her by holding games in her honor. Lugh carried out these wishes and the festivities and competitions that help to celebrate the harvest are named after him. Athletic competitions, music and dance, eating and drinking are all part of Lughnasadh.
Lugh was the god of all skills and arts and honored even today as the patron of blacksmiths.
The earliest mention of him describes him as a King of the Tuatha de Danann and the master of many arts. Later stories refer to him as the god of all skills and relate many stories about these skills. Still later, because of the harvest festival and games that were held at his order, he became associated primarily with the harvest.
His heritage, like many of the early Celtic gods, was complicated. Wikipedia does a good job of condensing it into a few lines:
Much of the early history of Ireland is recorded in the Book of Invasions, which recounts the many times Ireland was conquered by foreign enemies. According to this chronicle, Lugh was the grandson of one of the Fomorians, a monstrous race that were the enemy of the Tuatha de Danann. Lugh's grandfather, Balor of the Evil Eye, had been told he would be murdered by a grandson, so he imprisoned his only daughter in a cave. One of the Tuatha seduced her, and she gave birth to triplets. Balor drowned two of them, but Lugh survived and was raised by a smith. He later led the Tuatha in battle, and indeed killed Balor.
More information on Lugh can be found at Magic of Mythology
image found at PaganNews.com
Lammas, on the other hand, is a Christian celebration of the first wheat harvest. Farmers would take a loaf of fresh bread made from the current wheat harvest to church to offer as thanks and to ask for a blessing for the rest of the harvest. It was also called the Feast of the First Fruits and celebrated with bread and apples. This may a case of "if you can't beat em, join em" and a way to de-paganize a celebration without causing too much resistance. But this is primarily an Anglo-Saxon tradition and may only be related to Lughnasadh by coincidence. All farming cultures celebrate the harvest and these appear to be two separate celebrations by two different cultures at two different times in history.
I don't know how the two terms came to be interchangeable in neo-Pagan circles, but there is a distinct difference in their history.
Nowadays I think we are doing well to remember to celebrate the harvest at all (unless you happen to be a farmer) so I suppose the name is not all that important, but it is nice to know the origins of these things.
Celebrate by offering thanks for our abundant earth. Light candles of orange and yellow, burn incense of rose or sandalwood. Prepare a meal of lamb, wheat bread, apples and wine and offer it to Lugh or whichever deity you owe some thanks to. Don't forget to include some fun in the day including some games or competitions. photo found here