Help... I'm a lesbian in a man's body!!
Depending upon how your league rules are set up, when a player is dropped from any team, he goes to waivers for a certain number of days. In most leagues, this time frame is 2 days, or 48 hours. During that time period, any team who wants to make a claim for him may do so. The team with the highest waiver priority to submit a claim for him gets him. If nobody has submitted a claim after the waiver time has expired, then he goes to the free agent pool.
Your waiver priority should be set based upon the order in which you drafted. If you had first pick, then you should have last waiver priority, and if you had last pick then you should have first waiver priority.
When you make a successful waiver claim to pick up a player from waivers, your current waiver claim priority automatically gets bumped to last. This way teams who make waiver claims then have the lowest priority and future waiver claims from other teams will be needed to change their priority from last on up.
If you are talking about a Yahoo league, you can see the waiver priority of each team by clicking on the 'Team and Coach List' link on the right hand side of your options.
The biggest thing is if you have #12 waiver priority in a 12 team league, there is no worries about submitting a waiver claim because you can't get a worse waiver priority than you already have. If you have the #2 waiver priority though, and a player gets cut from another team that you have an interest in, you have to decide if it is worth using your waiver priority to pick him up, or sit it out and hope that he clears waivers and gets added to the free agent pool.
Any pick up from the Free Agent pool does not affect waiver priority. Only submitting a successful waiver claim will bump your waiver priority to last in your league.
Sounds about as clear as mud, doesn't it?