I was wondering what exactly "Private" meant in the title "Saving Private Ryan".
What was the significance of that word?
Private is a rank like a general, major, ...
A private is a soldier of the lowest military rank (equivalent to NATO Rank Grades OR-1 to OR-3 depending on the force served in).
Basically the title means that they are going to save a low ranked soldier named Ryan.
EDIT: Quora also offers the answer to your comment:
It emphasizes the thrust of the movie - that great human expense was made to rescue one 'lowly' soldier, because all his brothers had been killed.
This was largely covered in an earlier version of the film's script. The men (and indeed the audience and the title of the film) are all intended to question the relative worth of a group of men risking their lives to rescue an individual private in the Army, somewhere in France.
Miller: The numbers don't make sense, sir. His brothers are dead, that's too bad, but they're out of the equation. Sending men up there is bleeding heart crapola from three thousand miles away. One private is simply not worth a squad.
Colonel Anderson: This one is. He's worth a lot more than that. Which is why I'm sending you, you're the best field officer there is.
The irony of the film is that in saving Private Ryan (an individual soldier), the men of the company involve themselves in several key moments in the invasion of France, resulting in very many more men being saved.