Confusion about conventional current and electron flow and how it runs through and powers a circuit
Have you ever played this puzzle?  There are fifteen sliding pieces, numbered 1 to 15, arranged in a 4x4 grid.  It has one "hole" where there is no piece.  You can move any piece adjacent to the hole into that space.


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As you play the game, what is moving: the numbered pieces, or the hole?

Technically speaking, you are moving the numbered pieces.  They are the electrons, the physical objects that move around.  You can even get them to move in a closed path (a circuit).

But if you have played this game for enough time, you quickly realize how important the hole is to the strategy of the game.  A skilled player can move the hole anywhere he wants, and the hole can even move around in a closed circuit.  The hole represents the absence of electrons, and can move around just as the electrons themselves can.  In semiconductor theory, we even officially call them "holes" and treat them like a particle, even if they're not.

Whichever way you count, there is something moving.  We call this moving thing "charge".  Benjamin Franklin (as quoted below) called it "electrical fire".

Notice that whatever move you make, there are always 15 numbered pieces and 1 hole.  We say that this is conservation of charge, and was discovered by Franklin in the quote below.  As long as we are counting or adding things, it makes sense to make one kind of charge positive and another kind negative.  How shall we assign which is which?

The problem is that there has never been -- and never will be -- a device which can see electrons or holes.  They're just too small and too fast.  So Franklin arbitrarily assigned them:


  We suppose as aforesaid, That Electrical Fire is a common Element, of which every one of the three Persons abovementioned has his equal Share before any Operation is begun with the Tube. A who stands on Wax, and rubs the Tube, collects the Electrical Fire from himself into the Glass; and his Communication with the common Stock being cut off by the Wax, his Body is not again immediately supply’d. B, who stands upon Wax likewise, passing his Knuckle along near the Tube, receives the Fire which was collected by the Glass from A; and his Communication with the common Stock being likewise cutt off, he retains the additional Quantity received. to C, standing on the Floor, both appear to be electrised; for he having only the middle Quantity of Electrical Fire receives a Spark on approaching B, who has an over-quantity, but gives one to A, who has an under-quantity.
  
  If A and B touch each other, the Spark between them is stronger, because the Difference between them is greater. After such Touch, there is no Spark between either of them and C; because the Electrical Fire in all is reduced to the original Equality. If they touch while Electrising, the Equality is never destroyed, the Fire only circulating.
  
  Hence have arisen some new Terms among us. We say B (and other Bodies alike circumstanced) are electrised positively; A negatively: Or rather B is electrised plus and A minus. And we daily in our Experiments electrise Bodies plus or minus as we think proper. These Terms we may use till your Philosophers give us better.
  
  Benjamin Franklin, Letter to Peter Collinson, May 25, 1747.  Emphasis added.


Franklin wrote this letter to Peter Collinson, an English naturalist.  Franklin was somewhat right about "These Terms we may use till your Philosophers give us better."  Indeed, it was an Englishman (physicist J. J. Thomson) who discovered that the actual particles moving in a circuit -- which he called "electrons" -- are negatively-charged.  Unfortunately, the connotation of positive and negative stuck.