Elders mainly keep track of witches, watching from the Heavens. They communicate with those on Earth through their helpers and messengers, the Whitelighters. The Elders decide who may become a Whitelighter, and give and take their powers. A Whitelighter's ability to heal is channeled directly from the Elders.
It appears that most of the Elders were Whitelighters themselves at one time. Though some were probably witches because Kevin, a teenage witch, was promoted to be an Elder. Leo Wyatt is promoted to the rank of Elder after saving the Charmed Ones and the remaining Elders from the wrath of the Titans.
Though Whitelighters are not allowed to kill, the Elders have powerful fighting abilities. For the most part, the Elders' powers are only shown on screen when Leo is promoted to the rank of Elder. Leo is seen using a lightning-like power to kill Gideon. In one episode, three Elders are shown to kill Phoebe and Paige with magical flames, although this occurs in a magic-induced hallucination and does not indicate their actual abilities.
When witches need advice on demons they know nothing about, they can ask the Elders by sending their Whitelighter. The Elders are not all knowing, however; in some cases, they know no more than the sisters do. In one episode, Leo had to find out who the Four Horsemen were and what they were up to, because the Elders could not help. The Elders do not allow witches or any other magical being besides themselves and Whitelighters to enter the Heavens. Even the Charmed Ones do not have this privilege, as Piper and Leo have to ask for an audience when they seek permission to have a relationship (though this rule is trumped by Piper's god status), and Leo specifically tells Piper she is not allowed there when she comes to see him. The Elders do, however, allow Leo to watch over his son Wyatt and to take him to be with the Elders when he may be in danger.
As the show progresses, Paige begins using her Whitelighter side, and is able to converse with the Elders as she pleases. Though she does not go to the Heavens like the other Elders or Whitelighters do, they meet at the Golden Gate Bridge and she speaks with Sandra, one of the few female Elders.
Very few Elders ever come to Earth. Leo and Gideon are rare examples who spend most of their time outside the Heavens. A few come to Earth to speak directly about important issues, like Gideon's intentions to kill Wyatt and the possible threat of the Avatars. The Elder Sandra comes to speak to Piper about Leo during the last season. Overall, as a group, the Elders prefer to take a background role and remain largely unseen in the day-to-day affairs of even the magical world.
Thousands of years before the present, the Titans once ruled the world, but were abusive of their powers. Seeking to entomb them, the Elders of that time created the Greek gods by infusing mortals with a mystical power beyond their own. Ironically, the self-declared Gods of Mount Olympus would later force the world to worship them too for many years.