When I try to install nodemon i get this error message in node.js ?

Node.js error EACCES means you don't have access to the path it's specifying. In this case, /usr/local/lib/... is owned by the root user. See the command output here:

$ ls -l /usr/local/lib

total 12
drwxr-xr-x 4 root root  4096 Apr  7 23:45 node_modules

Nobody but root is allowed to write to the node_modules folder. This can be solved by prefixing your npm command with sudo to run the command as root.