For MX (from 3.10.1 onwards), the /web folder (when you unzip the distribution) contains some template files which you can optionally tailor (although any tailored file will get overwritten when you upgrade). However, the main purpose of the folder is to hold files created by the Cumulus code when you are running the software, ready for updating to your web site. These files contain data and have the file extension .json. Obviously, files created by the code cannot be tailored, but (depending on release) you may be able in Settings control whether these files are generated and whether they are uploaded.
Note that MX (from 3.10.1 onwards), uses web pages that are uploaded only once onto your web server. The data needed to populate most of these web pages is supplied by websitedata.json file in /web folder, which Cumulus will process and upload at regular interval.
In older releases of MX, the /web folder (when you unzip the distribution) contains many template files, these files contain a web page structure and represent data by web tags that MX will process. Some of these have filenames of pattern xxxxxT.htm. When you are running the Cumulus MX software, additional files are created in the /web folder, some of these will have .htm file extension, others will have .json file extension. Again some files can be tailored (although any tailored file will get overwritten when you upgrade), and some will be generated from internal code and cannot be tailored.
When you install the legacy Cumulus (up to version 1.9.4), a set of template files is included. As Steve Loft said, "They exist because these are our (Beth and Steve Loft) web pages, we never imagined people would use them without modification". Cumulus can process these templates (on a regular interval) and create web pages which it them uploads to your web server.
If you decide to use your own template files, perhaps as described here with PHP script, it is best if you store them in a different folder. More information on the Customised_templates page.