Someone with the user name "user" has logged in to this machine with host name "Linux-003".
"~" - represent the home folder of the user, conventionally it would be /home/user/, where "user" is the user name can be anything like /home/johnsmith.
"$" - this is a sign of the shell prompt, which means that the shell is ready to accept commands, you can understand it as a separator after which, you can interact with a shell. Can also be "#" which shows that root is the user whose session is going on.
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