English uses one word, compassion, for several things that Sanskrit and Pali keep apart. That is part of why the Dalai Lama can be hard to read in translation: the surface word covers several different ones underneath.
The four main ones are the brahmavihāras, the divine abodes, set out by Buddhaghosa in the Visuddhimagga and inherited by Theravāda and Mahāyāna both. Karuṇā, the same word in both languages, is compassion: the wish that beings be free from suffering. Maitrī in Sanskrit, mettā in Pali, is loving-kindness: the wish that beings be happy. It is a different cultivation from karuṇā, though English readers usually merge them. Muditā is sympathetic joy, gladness at someone else's happiness without measuring it against your own. Upekṣā in Sanskrit, upekkhā in Pali, is equanimity, the steadying ground that keeps the other three from drifting into sentimentality or burnout.
Bodhicitta, awakening-mind, is the larger Mahāyāna term. It is the aspiration to attain awakening for the sake of all beings. Karuṇā is one of its emotional grounds. Śāntideva's Bodhicaryāvatāra spends most of its length on what taking bodhicitta seriously involves, and the Dalai Lama's commentaries return to that text.
There is one distinction the Dalai Lama has drawn in conversations with Paul Ekman and Tania Singer that is useful for English readers: karuṇā versus what most of us mean by empathy. Empathy in the modern sense is feeling-with, registering someone else's distress in yourself. It is unstable on its own. It can tip into personal distress, and it can produce the fatigue anyone who has worked in care knows. Karuṇā is empathy plus the wish to relieve, and the wish is what keeps the feeling-with from consuming the person feeling it.
Thupten Jinpa writes about this in A Fearless Heart, and it is the framing behind the compassion training course he developed at Stanford. The English word does the work of several Sanskrit ones at once.