When teams want to clear roster space without immediately parting ways with the player they're removing from their roster, they can designate him for assignment. The move always clears a 40-man roster spot and when a player gets designated off of the active roster, a 25-man roster space opens up as well. The team then has ten days to assign the player to the minor leagues, trade him, or release him.
Players who have the right to refuse a minor league assignment benefit from the ten-day window, and it buys time for the team as well. For a team to assign a player to the minor leagues, he must first be exposed to all other teams on irrevocable outright waivers. If no one claims him, he can be released or outrighted to the minors.
A player's ability to refuse a minor league assignment depends on his service time and contract status.
Thanks to Rob Neyer's Transactions Primer and Cot's Baseball Contracts. Check out the MLBTR Glossary for more explanations of transactions terms.