I don't know the context in which you saw this phrase, so I can only answer this way. As I have seen it, the phrase, "papa lauae," has been used to refer to a particular hula class, either a graduating class or a current group of hula dancers.
"Papa" has many meanings and one of them is "class" such as a class of students. "Laua`e" is used to refer to two different sweet-scented ferns. It's a long story, but the laua`e name was originally used to refer to a native fern that has become rare today and the name laua`e is now used to refer to a common, introduced fern. Both have importance to hula because the common one is used in place of the rare one. This is a very good cultural practice, because the rare native one needs to be protected and not over-picked, and the common one is very easy to find and harvest. I will soon be putting up a web site on laua`e if you want to read more.
As for Papa Laua`e. In 1983, kumu hula Auntie Maiki Aiu Lake graduated a hula class and named it, "Papa Laua`e," the "laua`e class." This name refers to just ONE of her hula "graduating" classes. Other classes were papa `ilima, papa kukui, papa lehua -- to name just some of them. They are all named for plants.
Later, Kumu John Lake graduated a class from his halau, in 2003, and it was named Papa Laua`e, it was his (Kumu John Lake's) one "laua`e class." Recently, Kumu Hula Vicky Holt Takamine has started a group called Papa Laua`e o Makana. This is NOT a graduating class, but a halau, a hula school. In this case the name laua`e o Makana refers to the laua`e of Makana, an area in Kaua`i. There's more to it, but this answers your question, I think.
I am Puanani Anderson-Fung and I am a native Hawaiian ethnobotanist. I have been doing research on laua`e in Hawaiian language and culture and I hope to soon have a web site available within the next year (by the end of 2012) for people to read more about laua`e and its history in our culture. Today is 9/23/2011.
"Papa lauae" in Hawaiian refers to a type of seaweed known as Limu Lipoa. These seaweed strands are commonly found in coastal areas and are sometimes used in Hawaiian cuisine.
"Lā 'ou mai ke ola i ka papa hema ulu," in Hawaiian, translates to "I receive life from the southern realm." It is a traditional Hawaiian saying expressing gratitude for the gift of life from the southern region.
Crab in Hawaiian is called "puhi moana."
In Hawaiian, strawberries are called "ʻōhelo ʻai."
In Hawaiian, boogers would be referred to as "waiū."
Aloha: kaika papakema [kI ka papa kay ma]
Papa Honu Hoku Moku Enjoy
Bob = Papa
Papa malaa'o.
There is no native Hawaiian word for strawberry. Modern Hawaiian uses the term 'ÅŒhelo papa.
papa'i [papa ee]
In Hawaiian, strawberries are called "ʻōhelo ʻai."
Hoa Kula.
Crab in Hawaiian is called "puhi moana."
Aloha; You say papa. Shaka (bye)
In Hawaiian, boogers would be referred to as "waiū."
clam = Papaua [papa-oo-ah] or [papowah]
papa