What exactly does it mean when:
A player has taken assignment
A player has cleared waivers

Especially now that the real trade deadline is coming up at the end of the month.

Taken assignment= sent down to the minor's or be out of a job.
Cleared wavers= no team wants you, your free to find your own job.

I'll take a shot at this one.

A player, depending on time served in the Major Leagues, has a certain amount of times he can be optioned back to his minor league club. When that player is out of options, as was the case with Wily Mo Pena, the club can ask him to take an assignment but he's not obligated to take it. If he does take the assignment he simply reports to the minor league club. If he does not accept the assignment, the player goes through the waiver process and any team in MLB is free to claim him and the contract that comes with him. If no team is interested in signing him at all then that player "cleared waivers" and the team has a certain amount of time to either release the player or trade him, I think it's ten days.
If a certain team shows interest in signing the player in the waiver process, the current team has the right to pull him off waivers but they have to add him back to the major league roster.

That's the way I know it anyway.