One of the top reasons people don't get their priorites done, is that they simply have toooooo many.  This is part of life (both work and personal).  We love to over-allocate and rely on optimism and luck to get things done.

Humans are drawn to the number 3 and feel comfort with it.  It also aligns with a manageable number of priorities.

A system that we try to follow internally is that each employee no matter what their role is (CEO vs Field Labourer) must know their top 3 priorities for the day, week and month.  They must be able to achieve those priorities, the company depends on it (in today's world, everyone and everything is inter-connected).  Must remember, each role has a different level of complexity around their priorities, a different level of change control must also be followed and a different level of overlap with others exists based on the role - so it can be a communication challenge.  Internally, my company even has a "priority collaboration software" that helps to enable this (however is still a work in progress!!!).

Key challenges are to ensure that you are absolutely clear (you must know) what your own priorities are each day, week and month and that you are on the same page with your manager AND ensure you have a system to manage through the changes that occur.  Further description of this concept includes:

• Daily top 3 - these must be simple, fast and have a lower level of complexity/risk.  Scope of work is generally described in the format of tasks.  Try to put a measurement on them so they can be planned and you can track your progress, example:  you have approximately 7 hours available each day, 1 hour of the day is allocated to secondary priorities and admin, so ensure the 7 hours is realistic to get the priorities you take ownership over (you MUST try at all odds to get them done since they should build into the weekly priorities and also link to other people's priorities) 
• Weekly top 3 - these are more complex/risky but still must be clear enough in scope that the majority of the work is described in tangible, concrete work packages, think in terms that you now have 35 hours to accomplish the work so try to set realistic goals, then ensure to allocate the 5 other hours to secondary priorities and admin, and try VERY hard to not go over those 5 hours, cut off the work and defer it to the following week.  If your priorities change, then go through a mini-change control process.  One of your top 3 priorities must be demoted to a Secondary Priority, then new expectations must be established and you and your Sponsor/Managers/impacted stakeholders must all be on the same page.  Again, you must take ownership of the managing of these top 3, since is more tricky then the daily top 3, so ensure that you have more communication with your manager about them (are they on track, are they changing, what are the risks as we progress, etc)
• Monthly top 3 - these have the highest level of complexity and risk, and often can change over the course of the month.  However, they still must be planned and managed.  Ideally, they are documented, to help formalize them.  Ensure they are not too overly detailed, they can have aspects of intangible work, and thus can be further elaborated as the weeks are completed.  They can be thought of as guiding objectives that are detailed enough that you can "check them off" at the end of the month, but not too detailed that you have to spend days planning as to what they are.

A good mentor I worked with always said, confirm and then understand your top priorities, then at a minimum you must get them done, AND, fight off everything else and consider it as "white noise". 

Recently, I was able to relate this approach to an article I read (full article link), an excerpt of that article is below:

Set three specific goals weekly and monthly -- and ignore everything else.

There are a million things you can do to improve your business. The problem is, if you try to fix or improve them all at once, you end up not making much progress in any of them.
You need to be relentless in following your goals. At least once a day, I find myself asking, "Is what I’m working on directly helping one of the three goals?" It's easy to distract yourself with tasks that aren't critical. You just need to stay on top of it. This helps us accomplish big things quickly.
To hold ourselves accountable and stay on the same page, we have weekly meeting on Fridays. During these meetings, our company sets the goals for the upcoming week and see where we are on the previous week’s goals. It's a lot easier to say "what went wrong this week" then "what went wrong this quarter"
--Co-founder Patrick Ambron of BrandYourself

Related: The Best Business Advice You'll Ever Get