In anticipation of the many "how does the waiver wire work?" and "how do waiver trades work?" questions that will be asked this month, here is a nice article from Fangraphs.com summarizing the process. The basics are:
1. If a player clears waivers, the Royals can trade him to any team.
2. If a player is claimed on waivers, the Royals can either (a) work out a trade with the claiming team, (b) let the claiming team have the player for free (other than assuming that player's salary and contract), or (c) keep the player (which is called revoking the waivers). If more than one team claims the player, priority goes to the team with the worst record in the same league (AL), then to the team with the worst record in the other league (NL).
3. Only players on the 40-man roster have to go through waivers to be traded. Thus, if the Royals made a trade with another team, the Royals could only receive (a) minor leaguers not on the other team's 40-man roster or (b) players on the 40-man roster who have cleared waivers.
I like the note in the comments that the waiver process is now done by an email listing the players put on waivers that day -- as a kid, I imagined someone in each team's office sitting by a ticker tape machine reading off the player's names one-by-one.