nvm (Node Version Manager) is a tool that allows you to download and install Node.js. Check if you have it installed via nvm --version.
curl -o- https://raw.githubusercontent.com/creationix/nvm/v0.26.1/install.sh | bash
npm (Node Package Manager) is a tool that allows you to install javascript packages. Check if you have it installed via npm --version.
npm comes with Node.js so if you have node installed (node --version) you most likely have npm installed as well.
You don’t need nvm unless you you want to keep multiple versions of Node.js installed on your system or if you’d like to upgrade your current version.