Synodic month
The Moon takes about 29.53 days to cycle through its phases relative to the Sun.
Understanding lunar phases — Guide
Guidance for Moon Phase Calculator. Practical instructions, limitations, and accuracy tips.
This article explains the use and limitations of the interactive tool. No signup is required. Output can be approximate and should not substitute for authoritative sources when the stakes are high.
Waxing and waning
Waxing means the illuminated portion is increasing as observed from Earth; waning means it is decreasing.
Accuracy
Phase calculations here approximate the mean lunar cycle. Exact illumination and rise/set times require astronomical ephemerides.