npm install

Installs a package and all its dependencies.
Dependencies are driven by npm-shrinkwrap.json and package-lock.json (in that order).
without arguments: installs dependencies of a local module.
Can install global packages.
Will install any missing dependencies in node_modules.
It may write to package.json or package-lock.json.
When used with an argument (npm i packagename) it may write to package.json to add or update the dependency.
when used without arguments, (npm i) it may write to package-lock.json to lock down the version of some dependencies if they are not already in this file.

npm ci

Requires package-lock.json or npm-shrinkwrap.json to be present.
Throws an error if dependencies from these two files don't match package.json.
Removes node_modules and install all dependencies at once.
It never writes to package.json or package-lock.json.
Algorithm

While npm ci generates the entire dependency tree from package-lock.json or npm-shrinkwrap.json, npm install updates the contents of node_modules using the following algorithm:

load the existing node_modules tree from disk
clone the tree
fetch the package.json and assorted metadata and add it to the clone
walk the clone and add any missing dependencies
dependencies will be added as close to the top as is possible
without breaking any other modules
compare the original tree with the cloned tree and make a list of
actions to take to convert one to the other
execute all of the actions, deepest first
kinds of actions are install, update, remove and move.

The project must have an existing package-lock.json or npm-shrinkwrap.json.
If dependencies in the package lock do not match those in package.json, npm ci will exit with an error, instead of updating the package lock.
npm ci can only install entire projects at a time: individual dependencies cannot be added with this command.
If a node_modules is already present, it will be automatically removed before npm ci begins its install.
It will never write to package.json or any of the package-locks: installs are essentially frozen.