SUID (Set owner User ID up on execution) is a special type of file permissions given to a file. Normally in Linux/Unix when a program runs, it inherits access permissions from the logged in user. SUID is defined as giving temporary permissions to a user to run a program/file with the permissions of the file owner rather that the user who runs it. In simple words users will get file owner’s permissions as well as owner UID and GID when executing a file/program/command.
It can be applied in the following way
chmod u+s file.txt
Here owner permission execute bit is set to SUID with +s
chmod 4750 file1.txt
Here in 4750, four indicates SUID bit set, seven for full permissions for owner, five for read and execute permissions for group, and no permissions for others.
Example: passwd command
When we try to change our password we will use passwd command, which is owned by root. This passwd command file will try to edit some system config files such as /etc/passwd, /etc/shadow etc when we try to change our password. Some of these files cannot be opened or viewed by normal user only root user will have permissions. So if we try to remove SUID and give full permissions to this passwd command file it cannot open other files such as /etc/shadow file to update the changes and we will get permission denied error or some other error when tried to execute passwd command. So passwd command is set with SUID to give root user permissions to normal user so that it can update /etc/shadow and other files.