The "Others" are those Ancients who have ascended and follow their rules of non-interference. It has been implied that all "Others" may not be Ancients; when Dr. Daniel Jackson asked Oma Desala if The Others in the Astral Diner were Ancients, she responded "a couple of them" indicating that the group includes more than one ascended species (though she may also have been referring to the Others not being all of the Ancients). Indeed, there was at least one Asgard that had Ascended and become one of the Others thousands of years before and some humans, such as the People of the Cloister, have Ascended as well. Because of their fierce belief in non-intervention and free will, they have laid out somewhat cruel punishments towards those who break these rules, usually forcing them to watch the damage done as a result of their actions. For example, the Others forced Oma to watch Anubis use his ascended knowledge to take over the Milky Way and kill millions, while the planet of Velona was wiped out when they planned to use knowledge provided by Orlin to conquer other planets.
When Oma gave Daniel a second chance at ascension, all the Others did was watch her. They refused to even interact with Daniel and allowed Anubis to go ahead with his plan to destroy the life that they created. The only thing that caught their attention was when Oma sacrificed herself to stop Anubis as she'd finally learned her lesson. Given Daniel's return, the shutting down of the Stargate and the freezing of the self-destruct countdown, Daniel may have convinced them to intervene at least enough to send him back alive as Oma was no longer around to do it. Indeed, it was later stated that Daniel Ascended and returned a second time. However, the Others allowed Daniel to retain full memory this time. (SG1: "Threads") (SGA: "The Third Path")
Despite these harsh punishments, it is the Others' firm beliefs in free will and natural development that cause these actions. They feel that if they were to use their powers for even the best of intentions, it would make them no better than the Ori. While the rules generally do not apply if one of them returns to the lower planes before influencing events (such as when Orlin descended himself to help Earth manufacture a cure for the Ori plague), there have been exceptions to this, most notably when Merlin was developing a weapon capable of destroying ascended beings. In this case, they sent Morgan Le Fay to spy on Merlin and stop him if necessary. Indeed, Morgan did have to stop Merlin and destroy the weapon, but she preserved him in stasis in case the weapon was one day needed to be rebuilt. Whether or not the Others knew of this is unknown. Morgan tells Daniel at one point that the Others felt justified in their actions as Merlin returned to human form with much of his Ascended powers and knowledge intact which they felt was cheating the rules and was enough for them to interfere.
However, there have been times when The Others let things go at least somewhat before intervening. An example is when Morgan Le Fay helped Daniel and SG-1 find Merlin's weapon by appearing to them to guide them. She was disguised at first, but Daniel saw through her guise and The Others didn't intervene at first. They let her give out the Stargate addresses the team needed, but only intervened when she crossed the line and went to tell them exactly what they needed to know. Apparently enough of them agreed with her that they let her do that, but not enough that they let her cross the line completely. For that they banished her, but they did allow her to remain an Ascended Being, perhaps recognizing that she did what was necessary. They also didn't intervene when SG-1 rebuilt Merlin's weapon. SG-1 wasn't sure why as it was built with Merlin's ascended knowledge, but Daniel figured they may have wanted his plan to work and let it go. Adria believed that they wanted the Ori destroyed and allowed SG-1 to do it for them. Destroying the Ori meant saving themselves, and while they still refused to interfere in the war, Morgan believed that The Others owed the Tau'ri for saving them and intervened directly in the war. At first, all she did was send visions to Daniel which may not have been enough for them to stop her, but she later worked directly with the Tau'ri. The Others did not intervene, whether they were able to or not remains to be seen as Chaya Sar believed they could interfere with her actions across Galaxies, perhaps indicating that the Others supported Morgan Le Fay, indirectly, during these events. However, given Adria having all of the power of the Ori, they may simply have not been able to help without risking facing Adria. (SG1: "Ascension", "Threads", "The Fourth Horseman, Part 1", "The Fourth Horseman, Part 2", "Arthur's Mantle", "The Pegasus Project") (SGA: "Sanctuary")
While talking to the Vanir, Daniel suggests that perhaps the Others' rules are more flexible than they let on, as they don't intervene unless direct action is taken, such as allowing Oma to Ascend worthy others and allowing Morgan to give them the Stargate addresses. If an Ascended Being is not doing something directly but rather indirectly, they let it go. If they do something directly, then they interfere. Daniel suggests that as a result, the Others may be willing to allow the Vanir to receive help from Ascended Asgard Ran, as genetic material from her or information on how to repair damaged genetic material shouldn't be too bad. Indeed, when Dis approaches Ran for help, she indicates that the Others didn't immediately dismiss the idea, as they were discussing whether or not to follow the request rather than outright saying "no." What they would've decided is unknown as Ran chose to return to mortal form to help, seeing the possible extinction of her race as more important than the Ascended rules. (SGA: "The Third Path")