When a player's contract is designated for assignment -- often abbreviated "DFA" -- that player is immediately removed from his club's 40-man roster. Within seven days of the transaction (had been 10 days under the 2012-16 Collective Bargaining Agreement), the player can either be traded or placed on irrevocable outright waivers.
If the player is claimed off said waivers by another club, he is immediately added to that team's 40-man roster, at which point he can be optioned to the Minor Leagues (if he has Minor League options remaining) or assigned to his new team's 26-man roster (it was 25, prior to 2020). If the player clears waivers, he may be sent outright to the Minor Leagues or released. Players with more than three years of Major League service time or who have been previously outrighted may reject the outright assignment in favor of free agency.
Clubs may utilize this option to clear a spot on the 40-man roster -- typically with the intention of adding a newly acquired player (via trade or free agency), a Minor Leaguer or a player being activated from the 60-day injured list.