Padjal are a race of youthful horned beings in Eorzea among from which the Seedseers are born. They appear almost exactly like hyur except for their white (sometimes black-tipped) horns and the fact that they don't appear to age normally. Even the oldest of the Padjal appear hardly more than teenagers, in face as well as body shape and height.
Nothing much is known about the Padjal culturally. They appear to be a race of spiritual guides for the people of Gridania and live in the forests of the Black Shroud. They have the ability to communicate with the moogles and elementals of the forest freely, and can hear the voices of the aether and the crystals as well. They also appear to have high magical powers.
It appears that the current leader of the Padjal, or at least one of them, is Kan-E-Senna, who founded the current Order of the Twin Adder. It could well be that the single letter within Padjali names indicates rank of some kind.
All Padjal names are comprised of three words, the first two are the forename, and the final word is the surname.
The first name is constructed of simple vowel-based word (E, Ae, Uo, I, Ea, etc.) combined with a longer companion word. In the case of males, the vowel-based word comes first. In the case of females, it comes second.
Words in the Padjal language are normally separated by spaces. However, when a group of words forms a single concept (such as a proper noun, or a compound word), they are connected with hyphens.
In addition to names, the ancient language can be seen used in names of sacred locations around the Twelveswood. If a name is Padjali, you can assume that the location has been there for a long time.
The Thousand Maws of Toto-Rak (a beast from Padjali mythology)
Ak-Inik varlet -> servant of the Inik house
Ak-Mena varlet -> servant of the Mena house (Ak being the honorific prefix attached to a deceased member of a house)
There was also a question about how a Padjal gets their name (considering they start off as Hyur).
There are several old Hyur families in Gridania which consistently (albeit rarely) produce Padjals. Those include the Senna, the Yan, and the Pesi families. (Note the family names still possess the Padjali linguistic roots). When a baby is born to one of these families, he or she is given a "normal" Hyur name, as is the custom (i.e. Bryce Senna). When it is learned that the child is a Padjal, they are given a new first name based off the ancient Padjali tongue, and accepted into Stillglade Fane for their indoctrination.