Type 1 font is used in pdfTeX because of traditional TeX has the limit that each TeX font has at most 256 glyphs, while today most TTF or OTF fonts have more than that, so a mechanism of mapping is required to use these fonts, and due to the mapping was largely built on the Type 1 font concept, the limited TTF support in pdfTeX makes it more desirable to convert these files to Type 1 font first.
PostScript Type 2 is not an actual font format, it is a compression scheme. The same scheme is used in OpenType fonts but you won't yet need to worry about it.
Type 3 is font again, when raster font format was drawn in (and usually you won't like that to happen) you would see that, and you could use them with TeX if you are still targeting PostScript output. Most softwares would not support this, as it is something leftover in PDF standard.
Other numbers are either not actually font formats or only used for printer hardwares, and not related to TeX that much.
If you use XeTeX or LuaTeX that supports directly loading modern fonts from the system there are OTF versions of Computer Modern, which enables you editing them normally in Adobe programs. But after all Type 1 and Type 3 are just part of PostScript and as long as Illustrator keeps PostScript editing support you would still get albeit limited editing capability to them, like editing curves, change colors, it is only just these would not be treated as regular fonts so you would not able to retype them with the text editing tools in Illustrator.