One point is 1.3333 pixels at 96 dpi. The relationship is fixed by the CSS specification rather than by any physical screen: a CSS pixel is defined as 1/96 of an inch, and a point as 1/72, so the ratio is exactly 96/72.
About the units
The point is the traditional unit of type size, exactly 1/72 of an inch since PostScript standardised it in the 1980s. The pixel in web design is not a hardware pixel at all — the CSS specification defines a reference pixel as the visual angle of 1/96 inch at arm's length, and browsers then map that onto whatever the display actually has. On a high-density screen one CSS pixel may be two or three device pixels, which is why a 16-pixel font looks the same physical size on a phone and a desktop monitor despite vastly different pixel densities.
The exact factor
The factor is exactly 96 ÷ 72 = 4/3 = 1.3333. So 12 pt is 16 px, 9 pt is 12 px, and 18 pt is 24 px. The 96 dpi figure comes from Microsoft's decision in Windows to assume that density for screen rendering, which the CSS specification later adopted as the reference. It was never true of any particular monitor; it was a convention chosen so that text specified in points would come out at a readable size on a typical 1990s display.
Where you meet this conversion
Web and interface design, where a designer works in points in a print-oriented tool and a developer implements in pixels. Style guides that specify type in points have to be translated, and 12 pt body text becoming 16 px is the single most common conversion in the field. Print-to-web adaptation of brand guidelines runs through it. Note that modern CSS practice favours rem units over both, precisely because they scale with the user's chosen base size rather than being fixed.