Literally, karuna means "to mourn", "to pity", "to lament" - but the Buddhist meaning is quite different.
In Pali Canon karuna is one of the four brahmaviharas, practiced as an antidote against aversion towards people and society due to their shortcomings. Somewhere in the Canon I've read Buddha saying karuna is like what a good king should feel for his people. The king does not complain that his country is imperfect nor does he resent his people! Instead, he appreciates their condition and tries to help them out as much as he can. (This doesn't mean he lets them sleep in his bedroom - he can't solve their problems for them, but he can "give them the fish pole and teach them how to fish".) There is a reason they are called "brahma-vihara"-s, the dwellings of Brahma - the nondualistic all-inclusive God-Absolute. It is like the practitioner takes on the Brahma's perspective.
In Mahayana compassion goes hand in hand with wisdom and skillful means, exemplified by Lotus Sutra's parable of the burning house. It is like compassion of parents towards their children. They know there is sex, but they don't want their children to be harmed by what those are not ready for, yet. They know Santa Claus is an illusion, but because they love their children so much they put the gifts under the Christmas tree.
As you see, in both cases it is a top-down compassion, compassion of superior taking care of the junior, weak, sick, or confused - using whatever tools you have at your disposal, from logic, to useful fiction, to wrath and "tough love". Instead of resenting the way the world is or the people are (=the victim stance), we take the master's stance.
At the same time, part of the practice of deconstructing the ego is to cultivate clear understanding that in principle, there is nothing substantial that makes Bodhisattva "better" than others, beside the luck of having encountered, and benefited from, the genuine teaching. Bodhisattva can relate to everybody's dukkha - because they see that deep inside people are not that different. Everyone is subject to the three marks of existence. Every one is subject to emptiness. Even though Bodhisattva has realized Emptiness, s\he sees how hard it is to not assume things to be real. Bodhisattva can see how people organize their lives around illusions - and how it inevitably leads to suffering. This is why it is said, in one who sees things as they are, compassion arises spontaneously.
So [in my understanding] Buddhist compassion is not as much just feeling other people's pain -- as it is an attitude, a modus operandi permeated with the sense of ownership of and responsibility for the good of all beings - based on being able to relate, to put oneself in their shoes.
Several quotes to support my understanding.
Thich Nhat Hanh:
[Karuna is] the intention and capacity to relieve and transform suffering and lighten sorrows. Karuna is usually translated as 'compassion', but that is not exactly correct.
1) When there is suffering in others it makes (karoti) good people’s hearts tremble (kampana), thus it is compassion (karuna).
2) It combats (kinati) others’ suffering, attacks and demolishes it, thus it is karuna.
3) or it is dispersed (kiriyati) over the suffering, is spread out through pervasion, thus it is karuna.
As for this great compassion (maha-karuna), reverend Saradvatiputra, it means "work" or "action" (kar-)
[Even if is for the sake of others] it is one's own work - thus it is called "great compassion".