Ye olde
Old English had the thorn character, þ, capital: Þ, pronounced as a 'th' sound, from old Norse influence.
These days, only Icelandic still uses it.
Paper was expensive, so space was somewhat at a premium, so making common words like 'the' shorter was a decent idea,
which seems the main reason that printers would often opt to typeset it differently.
They might have made a þͤ, but when printing came about, the thorn was already in decline, so it didn't really make sense to create a new, standard, distinct piece of metal type.
It seems that the reason that printing would often typeset it using a Y with an often-superscript e is that the thorn in decorative script looked a bunch like the (blackletter) Y, like 
.
So Ye became an alternative used in printing for the word "the", understood as exactly the same thing, just a shorthand.
They also used this pseudo-thorn for the few abbreviations of other common words starting with th, like this and that(verify).
And þͤ is hard to type, so we use ye or just ye to refer to this printing practice.
Yet today, when you "Ye olde shop" on a sign, people will easily pronounce the y as the vowel (yee) and not 'the'.
Which makes sense because both the thorn and this printing-press convention are long gone.
The leftover issue is that a completely separate 'ye' also existed at the time, as a plural pronoun (similar to modern american y'all), as in e.g. hear ye, as you may remember if you've read some Shakespeare.
And both ye and ye, if used today, will be old timey references,
but when one means "y'all" and the other means "the", that's... confusing.
And incorrect in terms of said old times and now - ye olde shoope is either a mispronounciation the thorn, or it's gramatically incorrect use of the plural pronoun ("Y'all shop").
Meaning we have actually invented a third thing,
that we pretend is old,
but was never really part of the language,
just a thing we put on modern tourist traps.
Sure, it does basically mean that now.
That's what language change is.
Its history just amuses me.
See also: