A very small number of the Companions are Grove-Born, which means that they were called to materialize in the Sacred Grove in Companion's Field. While natural-born Companions tend to be reincarnations of Heralds and other Companions, the Grove-Born were never human and are probably true avatars.
The first of the Grove-Born were called by King Valdemar himself, who had been praying in the Grove for a way to keep the Monarchs of Valdemar from being corrupted, and to keep his people safe for all time. Given that he was also a mage, there was probably some spellcasting involved as well. Three Companions emerged in answer, bonding to King Valdemar, his son Restil, and his Herald Beltran, beginning the Heralds of Valdemar.
Since then, most Companions are born from other Companions, as most natural creatures are. A very select few, however, are born (or reborn) at the Sacred Grove as needed. In general, these Companions come with a few extra skill sets. This usually includes the ability to Mindspeak over longer distances than normal.
The Companion of the Monarch's Own Herald is always Grove-Born. These Companions essentially take turns serving together with the Monarch's Own, being reincarnated repeatedly. In addition, the mare Gwena emerged from the Grove to be the Companion to Elspeth, the first Herald-Mage since Vanyel.