With the Brahma Viharas, the Pali words are Metta, Karuna, Mudita and Upekkha. Now, I'm going to go from the backend. Upekkha means Equanimity, that is agreed upon by almost all teachers and scholars, Upekkha means Equanimity. Mudita means Sympathetic Joy, there are slightly different words that they use for Mudita, but it always basically means the same thing, having Sympathetic Joy with good actions and good qualities and so on. Karuna means Compassion by itself. Everyone agrees with these three in their basic sense, that Karuna means Compassion by itself.
However, there is disagreement over the translation of Metta. We translate Metta as Compassion/Lovingkindness, putting the two together. There is a wish for no pain and there is a wish for happiness. Now, Karuna being Compassion by itself means there is the wish for no pain. That is actually a good enough wish for the whole world, isn't it? Just the wish that there was no more Dukkha on the whole planet, that would be a nice enough wish. But Metta is bigger than that, Metta adds the wish for happiness, for peace of mind, for ultimate freedom and so on.
Thai people tend to use the word Metta in this way, including both Compassion and Lovingkindness. Some Western teachers are saying that Metta only means Lovingkindness, implying that it only is the happy part, wishing for happiness. So their Metta is very different to our Metta. There are some other teachers who will say Metta means Lovingkindness, but when they translate their Lovingkindness it actually includes the Compassion within it. So in this day and age of Buddhism in the West it is a little confusing as to the word Metta.
But as we translate it, it is both Compassion and Lovingkindness together. That is why we are continually saying that we are doing a Compassion/Lovingkindness meditation. Some people who want to use a Pali word for it, they would say we are doing a Metta meditation. But that is where you have to be careful, because somebody else's Metta meditation might be totally different to ours, because they don't touch on the Compassion part.