In Baseball, what does it mean when a player is "designated for assignment"?

When a player is designated for assignment, he is immediately removed from the club’s 40-man roster. This gives the club 10 days to decide what to do with the player while freeing up a roster spot for another transaction, if needed.

What happens after the 10 days? That’s a deadline by which they have to trade him, send to the minors, or put him back on the roster?

It depends they might trade him, send him, or he goes back to the roster.
Options:
Place the player on waivers (which can only be done within the first 7 days of the 10-day period)
Trade the player
Release the player

Thanks!

You welcome :)

Thanks! I was wondering too!

Great question!!