The definitive meaning of Designated for assignment:
Any player on the 40-man roster may be designated for assignment. The term is sometimes abbreviated as DFA or DFA'ed. When a player is DFA'ed, he is immediately removed fromt the 40-man roster.
The player's team has 10 days to from the date of designation to return the player to the 40-man roster or to trade, release, or outright the player to Minors. A player may only be designated for assignment if the 40-man roster is full and a player needs to be added.
However, in order to outright a player, he must first clear outright waivers. While the player is on outright waivers, he may be claimed by any other team for the $20,000 waiver fee. Even further, if a player is outrighted and it is his second removal from the 40-man roster during his career, then he has the option to decline the outright assignment and opt for or declare free agency instead.
Also, if player is released, then the team must pay all of the remaining salary unless another team signs him--which any team can do for as low as a prorated portion of the 327K minimum salary--in which case the original team would only pay the remaining difference. (In other words, the player's new salary is subtracted from what the original team still owes him.) Additionally, before a player can become a free agent, the team must ask for release waivers on the player. While he is on release waivers, any team claim the player for a $1 waiver fee, but that team also would be responsible for all of the player's remaining salary.
The only time a designated player can be returned to the 40-man roster is when the player provides proof of an injury that hindered performance, in which case the team may place that player on the disabled list.