For this particular case following code means:
Somebody with user name "user" has logged in to the 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.
"$" - is just a sign of the shell prompt, means that 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 who's session is going on.