If you choose to display the Lunar Calendar (by clicking "Lunar" in the Fixed Menu at the top of the page), it will show as a black area with silver outset borders, at the top-left of of the main window of this site.
The Lunar Calendar shows images of the 4 main phases (quarters) of the moon (New, First Quarter, Full & Last Quarter) organised in a circle (obviously, the New Moon image is black-on-black, and thus invisible). The images rotate anti-clockwise as the lunar month progresses. The vertical blue line with in-pointing arrows indicates the current (now) point in the lunar month.
Clicking (once) in the Lunar Calendar shows this page.
Moving your mouse over an image of a moon phase shows a pop-up of the dates and times of the previous and next times that phase happened or will happen. These pop-ups also show the current date & time in UTC (basically, GMT) and in your local time (based on your PC's locale setting), and an image of the moon-phase about which you requested data.
The Lunar Calendar reloads every minute, to ensure that the position of the images in the Lunar Calendar are up to date. The pop-ups are also updated every minute.
The dates & times of the quarters of the moon are calculated for viewing from Greenwich (London). They may vary by a few minutes for other locations. Since all dates & times of the phases of the moon are in UTC, you will need to adjust for your local time zone in order to know the times for your locale. You can work out the time zone difference between UTC & your local time by comparing the values for the current date & time in UTC and in your local time, on the last 2 lines of the pop-ups.
If you want to see the dates and times of the phases of the moon, adjusted for your location, and in your local time zone, try this site. You can enter your location where it says "Place or country", and a select list will pop up (only major cities - click to select); then the times will be correct for your location.