What it means
Only three names appear in it: Hare, Krishna and Rama. Hare is usually read as a call to Radha, the Lord's own energy. Put together, the sense is something like: O Radha, O Krishna, O Rama, take me into your service. There are sixteen words in all, which is why it's also called the sixteen-name mantra.
Where it comes from
The mantra is found in the Kali-Santarana Upanishad, where Brahma tells Narada that it cuts through the faults of Kali Yuga. The Upanishad gives it with the Rama half first. In the sixteenth century Chaitanya Mahaprabhu took it into the streets of Bengal and Odisha as kirtan, and the order most people know today comes from his followers. ISKCON later carried it around the world. What people love about it is that it asks for nothing: no initiation, no ritual, no special hour.
How to chant it
A tulsi mala helps. Start at the bead next to the Sumeru, say the whole mantra once per bead, and move the beads along with your thumb and middle finger. When you reach the end, don't cross the Sumeru; turn the mala around and go back. Say the words clearly enough to hear them yourself. Listening is most of the practice.
When: Early morning is best, but this mantra carries no restriction of time or place.
Try it now
The japa counter opens with this mantra selected and keeps count for you.