There are two types of waiver order depending on the types of player acquisition system the league uses. They have different names but behave similarly.
In this FAQ we'll go through the similarities and difference between them.
Waiver Order
Free Agent Budget Tiebreaker
Similarities
Differences
Waiver order in Standard Waiver leagues: under "Players" -> "Waiver order"
In public fantasy leagues, ESPN is the league manager thus no one in the league can edit the order.
In Private leagues, LM has the ability to edit such order. Simply click "Edit" and use the drag function which should appear on the right. LM will need to make sure to click "Save changes" to finalize any made changes. 
Free Agent Budget Tiebreaker in FAB leagues: under "Players" -> "Budget Summary"
In Private leagues, LM has the ability to edit each teams budget. Simply click "Edit" and edit the amount shown for each team. LM will need to make sure to click "Save changes" to finalize any made changes.
Similarities:
1. How it changes -Both systems have two ways for the list to refreshes and LM can decide to change it at anytime.
Reset Each Week -> Each Tuesday morning at 12am PT/3am ET when new fantasy week starts, the order resets to inverse standings.
Move to Last After Claim -> Once a manager successfully claims a player via Waiver process (DIFFERENT THAN FREE AGENCY PICKUP!), that manager will move to the bottom of the order. This means this list NEVER resets.
Please note: Regardless of which option is selected, after a team makes a waiver selection, that team is shifted to the end of the bottom of the waiver position.
Please note: If any of the league settings are changed/updated, waiver order will reset to inverse order of standings at that time.
2. Initial order -
Both system has the same rule with how initial order are set.
If you play in a Salary Cap league, the team with the most money remaining after the Salary Cap draft is complete will get the most favorable waiver position. If two or more teams are tied with the amount of money they have remaining, then the team that filled out their roster first during the draft will get the more favorable waiver position.
If you play in a snake draft league, the team that had 1st pick would be last in the order. Last Pick would get the most favorable position. You get the idea! (INVERSE ORDER OF DRAFT ORDER!)
Differences:
LM can edit waiver order but CANNOT edit budget tiebreakers. meaning once a waiver is processed and a manager is moved to bottom, LM cannot change the order. (LM can reverse the claim, not the order)
2. What transaction affects it? -
In leagues that use "No Waivers", "Waivers" or "Free Agent Budget", please note that FREE AGENCY PICKUP does NOT affect the waiver order/budget tiebreakers
This is a Waiver Claim:
