As Aristotle might remind us, “love” is said in many ways. Moreover, there are at least four separate Greek words with distinct but overlapping meanings that we translate by “love” — eros, agape, philia, and storge.
Eros most commonly emphasizes passion, sensuality, and attraction. Classical authors often associated it with a kind of mania leading lovers to extreme behavior. Modern authors have generalized it to include desire of all sorts, and Freud in his later work treated it as a sort of life force. Plato in the Symposium and Plotinus in his works on Beauty and Intelligible Beauty saw eros as capable of being sublimated into an uplifting kind of love for ideal or spiritual things. Aristotle poetically gave it a cosmic role, saying that the stars are moved by eros for their apparent axis of rotation. The latter, as cosmic “unmoved mover”, “unmovingly moves” things in this way, by being the object of their eros. (Unmoved moving also has another, purely descriptive sense that is not relevant here; see Moved, Unmoved.)
Agape is the main word for love in the Greek New Testament, emphasizing compassion and charity. It is applied to God’s love for the world, and in the injunction to love our neighbors as ourselves. It is about this kind of love that Augustine said “love, and do as you will”.
Philia is applied by Aristotle to a wide range of ethical and social contexts — a feeling of affection and sympathy between friends, lovers, families, members of a community, people engaged in some common activity. In the Rhetoric, he defines it as wanting what we think is good for someone, not for our own sake but for theirs, and being inclined to act on that insofar as we are capable. It involves an implicit norm of reciprocity in a broad “proportional” sense that applies even when there is some asymmetry in the underlying relationship. Aristotle argues that although a kind of self-sufficiency is also a virtue, doing for others is a greater good. Moreover, he says that the philos (friend or loved one) is for us like another self. This is the Aristotelian root of Hegel’s ethics of mutual recognition. Also, philosophy is philia for wisdom.
According to Wikipedia, storge is familial or domestic love. Modern authors have associated it with long-term commitment and a kind of unconditional support, and with romantic love that has origins in friendship rather than manic attraction.