What does it mean when a MLB player is out of options?

Answer 1

That means that a player, who is on a team's 40 man roster, may not be moved to the minors without first clearing waivers.

MLB teams are limited on how many times they may move a player on their 40 man roster to the minors. Each time a team does this and the player spends at least 20 consecutive days in the minors, the team uses one of the player's 'options'. Only one option may be used per season so a team may move a player from the minors to the majors and back many times during a season with only one option being used.

A team has three options with a player. Once those three options have been used, the player is 'out of options' and must be placed on waivers and pass through without being claimed by another team to be moved to the minors.