You already discovered the answer for your monitor. I'll leave a general answer for others with a similar question.
The Video Electronics Standards Association has a set of standards for mounting flat panel displays like TVs and computer monitors. It is usually referred to as simply the VESA Mounting Standard. Pretty much any modern display that has built-in mounting holes complies with this standard.
The standard defines a hole pattern (usually rectangular, but larger sizes can have more than four holes and there can be a center hole), and screw specification (the screws are metric). The best explanation I found of the details (outside of the actual standards), is this Wikipedia article: https://en.wikipedia.org/wiki/Flat_Display_Mounting_Interface
The hole pattern and screw specification are based on the display's size and weight. If you see a rectangular hole pattern and the measurement is consistent with the standard for a display of the size and weight of yours, it's a safe bet that its a VESA mount. A quick summary of mount dimensions to check against:
(Click on image for more readable size) courtesy http://www.ergotron.com/en-us/support/vesa-standard
A more detailed table, including screw specifications, from Wikipedia:
- If a screen is heavier or larger than specified in table 1, it should use a larger variant from the table, for instance a 30-in LCD TV weighing more than 50 lb (23 kg) would need to use a part F mount.
- The weight limits were unusually chosen as round numbers in kg or lb for different sizes.
- The screw lengths for part C, D and E become whole numbers when adding a 2.6 mm thick bracket (which is how the standard describes them).
The screw lengths for part F are minimum / maximum / hole maximum, as in: M6 screws must go at least 9mm in but at most 10mm in, and the hole might not be deeper than 12mm.
The minimum length is important for strength. The maximum length is also important. You don't want the screw to force its way into the inside or to bottom out in the threaded insert so that it can't be properly tightened. Some mounts come with screws that are longer than necessary and then use spacers behind the mount to correct the effective length. If you use one of those screws as a guide at the hardware store, make sure you have any needed spacers.
There is additional detail, especially for larger sizes, in the Wikipedia article.
The VESA mount screws are likely to be different from those used to attach a stand for desktop or tabletop use. Note that, particularly for larger displays, the display manufacturer is allowed to choose screws different from the standard. If they do, they must provide the screws with the display, in which case you need to use the display manufacturer's screws in lieu of those supplied by the mount manufacturer and should have a sample.
If the display came with screws and you no longer have them, you might find a reference to their size in the product documentation or might be able to find the information online (or check with the manufacturer's customer support). Otherwise, it's trial and error for the thread (for an initial estimate, you can compare the hole diameter of the threaded insert to a chart of screw dimensions). I would refer to the VESA spec for the screw length, and verify that the threaded hole accommodates that length.