The physical appearance of the Eldunarya would vary through size and colour, relating to the colour of the dragon, and its age at the time of disgorgement.
Usage by Dragons
Eldunarí could be used by dragons as means of storing their consciousness, allowing them to "live" after their bodies had failed them and also becoming a powerful tool of communication while the dragon remained alive. Typically, if a dragon didn't store their consciousness in it, a dragon's Eldunarí slowly dissolved along with the dragon's body after it died. However, if a dragon wanted to, it could place its consciousness into its Eldunarí, turning it the color of the dragon's scales and making it glow. This would make the dragon's consciousness live on forever within the Jewel, unless the Eldunarí was broken. A dragon whose body had died, would live on inside his or her Eldunarí. However, a dragon could not smash its Eldunarí, so if it wanted to follow its Rider into the void (as some did) then it would either get someone else to smash their Eldunarí for them, or destroy it by one of the dragons' inexplicable pieces of Magic. The Eldunarí could also provide others with their own energy. An example of this is how Galbatorix used the Eldunarí of dragons to boost his power and take control of Alagaësia.
After a dragon had disgorged its Eldunarí, anyone in contact with it could call upon the dragon's strength and communicate with them over great distances as if they were standing next to the possessor. The energy an Eldunarí obtains for their sustenance comes from magic. When a dragon becomes resigned to their heart of hearts after Death, they take with them the amount of spare energy they had when they died. Over five to seven years after the dragon's death, the amount of energy the dragon is able to store increases to astounding proportions. Theoretically they obtained this energy through absorption of ambient energy or light.
Glaedr mentioned that when a dragon had died but had disgorged its heart of hearts and the Rider lived on, the pair became known as "Indlvarn". Although it may have been difficult for the dragon, there were many Riders and dragons that adapted and continued to serve the Riders. Often, when the Rider of the Indlvarn pair died, the trauma of the loss would drive the dragon to insanity, and the dragon contained within their Eldunarí would sometimes arrange for someone to destroy it for them.
Once a dragon disgorged its heart of hearts, it would not grow any larger than it was at the time of removal. It also couldn't be restored to its position in the dragon's body.
Increased Mental capacity
The Eldunarí can increase the Mental powers of the user, or not if the user is mentally incapable. They can increase both the range and mental strength of the user to both be able to withstand and attack the minds of others, almost enslaving the dragon's mind to there will, this could drive them mad (Eldunarí were known to go mad through extreme age as well). The increase is proportional to the size of the Eldunarí at the time of disgorgement. With his stash, Galbatorix was able to reach out with his own mind across all of Alagaësia.
Glaedr also told Eragon that the larger an Eldunarí was, the more energy it could store (which in turn means more powerful gramarye spells), which would provide incentive for Riders to wait until their dragons were old before having them disgorge their heart of hearts.
One benefit to having one's dragon disgorge their Eldunarí was easy communication between dragon and Rider over long distances.
Multiple Eldunarí were capable of communicating with each other if they were close enough.
Eldunarí in the books
The last remaining Ra'zac told Eragon "He (Galbatorix) has more heartsss than you do," which Eragon later realized was likely a hint regarding Galbatorix's Eldunarí.
After Murtagh asked if Eragon would be willing to go peacefully to Galbatorix, Eragon responded by saying "I would sooner tear out my own heart!" Murtagh replied "Better to tear out my hearts," a comment Eragon again realized in looking back was almost certainly a reference to Eldunarí.
Glaedr disgorged his Eldunarí and gave it to Eragon and Saphira before they all left Ellesméra. It was later revealed to Eragon and Saphira that both Galbatorix and Murtagh drew their incredible magical prowess through the enslavement of many Eldunarí. Barst, an important general in the Imperial army had also kept an Eldunarí, cleverly concealed under his breastplate, acting as a fake belly, which is why his strength and speed were equal to an elf's.
When Glaedr revealed to Eragon the history of the Eldunarí, he mentioned that before making their pact with the Elves, the Wild Dragons kept their Eldunarí in Du Fells Nángoröth, the mountains in the center of the Hadarac Desert.
Eldunari that escaped Galbatorix's control during the fall were hidden on Vroengard, in the Vault of Souls. These Eldunari helped Eragon cast the Empathy Spell, which defeated Galbatorix. Their existence was hidden by a powerful magic that erased the memory of the location and the name of the Rock of Kuthian that served as the entrance from all who knew of it, or who read it. The power of the enchantment and the cleverness of the concealment were such that Glaedr was fully convinced that was impossible that there could be any hidden stash of Eldunarí anywhere within Alagaësia.
The name "Eldunarí" almost definitely derives from the Old Norse word, "Aldrnari," one of several words for "fire".
The Eldunarí could be seen similarly to an airplane's "black box".