Ahimsa traditionally meant "do not kill or hurt people." This can be extrapolated to mean that we should not be violent in feelings, thoughts, words, or actions. At root, ahimsa means maintaining compassion towards yourself and others. It means being kind and treating all things with care.
Satya means "truth," or "not lying." Practicing satya means being truthful in our feelings, thoughts, and words, and deeds. It means being honest with ourselves and with others.
Asteya, or "not stealing," refers to the stealing that grows from believing we cannot create what we need. We steal because we misperceive the universe as lacking abundance or we think that there is not enough for everyone and that we will not receive in proportion to our giving. Because of this, asteya does not only consist of "not stealing," but also of rooting out the subconscious beliefs of lack and scarcity that cause greed and hoarding in all their various manifestations.
We practice brahmacharya when we consciously choose to use our life force (especially the energy of sexuality) to express our dharma, rather than to frivolously dissipate it in an endless pursuit of fleeting pleasures. Brahmacharya reminds us that our life force is both limited and precious, and sexual activity is one of the quickest ways to deplete it. As yogis, we choose to use the power behind sexuality to create, to fulfill our mission, to find and joyously express our inner selves. The practice of brahmacharya is not some archaic form of moralizing, but rather a reminder that, if we use our energy wisely, we possess the resources to live a fulfilling life.
Aparigraha means not coveting what isn't ours. It is different from asteya, which asks us to avoid stealing that is motivated by a greed springing from a perceived lack of abundance. Aparigraha is the greed that is rooted in jealousy. The Mother used to say, "Jealousy is a poison that is fatal to the soul." Jealousy means that we desire to be what someone else is, or to have what someone else has. Rather than finding who we are, we look at someone else and say, "I want to be that." Aparigraha, in its essence, helps us discover our own selves so that we no longer feel the need to covet what someone else has, or be what someone else is.