What do the various values of the Ligatures= option of fontspec do?

The fontspec package basically just activates certain features available in the font. The options differ depending on what type of font it is.

For OpenType fonts

The following table shows which options are available and which features they load (from the latest version of the fontspec package documentation, fontspec.pdf, 2020/02/21, § 3.1.8 with table 11 on p. 41):

What these do in the context of a specific font depends entirely on how the font designer set up that font. I would assume that fonts which make heavy use of such features come with a documentation that explains which ligatures are available and through which feature they are accessible. If not, you can inspect the font yourself using font editing software like FontForge or Glyphsapp (this will require that you can read OpenType feature code instructions).

If you want to know more about the features rlig, liga, clig etc., how they are intended to be used, how they differ from each other etc., you will find that information in the OpenType spec on the Microsoft website.

Note that the option TeX above is different from the others, as you already seem to be aware (and user @cfr has also mentioned). A feature tlig does not exist in the OpenType spec. This is something TeX-specific to emulate TeX's behaviour for interpreting certain ASCII input.

For AAT-fonts (Apple)

For AAT-fonts, fontspec accepts the following Ligatures options (§ 3.1 in the documentation):

Required, Common, Rare (or Discretionary), Logos, Rebus, Diphthong, Squared,
AbbrevSquared, and Icelandic

Some information on what these do is available in the section Font Feature Registry in the AAT specification on Apple Developer. Again, it depends on the individual font what will actually happen when you activate these features. (Some of these are discouraged from use, e.g. Logos; Icelandic does not seem to be mentioned in the spec).