To elaborate on Muke's contribution,
The Latin "hymn" from which the notes are derived are actually taken from a prayer (not quite the same) to St. John. The notation from said prayer was taken by Guido d'Arezzo some time in the 11th century, as a way to notate and preserve music based on Pythagoras' original conception and construction of the musical scale. 'Do' is not merely a nominal mistranslation - it was changed from "Ut queant laxis" to "Domine" by the Italians for some cultural, indiscernible reason.
To call this a "traditional" way to sing the scale is a very Western-biased claim, but mostly true in the context of the Classical Western Music tradition. The 'do re so' system, in theory and in practice, is known as Solfege. Today its widest and most practical application is for vocalists or vocal ensembles to sight-sing melodies. It is an easy and comfortable way for musicians of
all most skill levels to learn how to sing a piece by rote.
By choosing a tonic note (Your starting point upon which your scale is based - 'do'. Think of it as 'home' or whatever.) and singing the 'do re mi...' scale up to an octave (in the correct intervals, of course) you are constructing a Major Scale in the Ionian Mode, the most fundamental and important type of scale. Muke's WU also assumes that C major is the only major scale that exists, but it is only 1 of 12 Major Scales that theoretically exist, for the 7 natural notes and 5 accidental notes in Western theory (not including enharmonics, which, well, don't worry about them if you don't have to).
When studying a Solfege piece in a Major key in Ionian mode, you'll often run across notes outside the Diatonic collection (these are not necessarily accidental notes. For example, if you're in (fucking) F# Major, all the notes in the scale are accidentals, and all chromaticism you run across will be from naturalized notes (or double sharps I guess, God save the queen)). So there's a sub-system in Solfege used for raised and lowered notes in the scale that modifies the syllables:
^ di ri --- fi si li -- di
- do re mi fa sol la ti do
v -- ra me -- se le te --
Missing notes from these systems, such as a lowered Fa or a raised Ti do exist, theoretically, but are so dumb and impractical they're not named or used. The existence of "Si" as a raised 5th scale degree is a very likely candidate among reasons why "Si" did not survive as the 7th scale degree from the original Latin prayer. But I'm not sure why it was replaced by "Ti" or if there was a specific reason at all.
To construct a scale in a Minor Key using one of the three Minor Modes, just modify a few of the notes in a Major Scale on Solfege.
Natural Minor Mode:
do re me fa sol le te do
Harmonic Minor Mode
do re me fa sol le ti do
The Melodic Minor scale is the tricky one. It raises scale degrees from the Natural Minor mode as the scale ascends, then returns them to the values of the Natural Minor Mode as the scale descends (I am intentionally careful of my word choice - to say 'raise it then lower it again' is dangerous, and will not apply to all instances of scale construction).
Melodic Minor Mode
do re me fa sol la ti do te le sol fa ...
Most music you'll run across in a minor key will be constructed in Harmonic minor, because Melodic Minor feels graty and awkward, and Leading Tones are important. Very important. That's why even when you're working with what's mostly Natural Minor mode, your V(5) chord is still seen as cadencial - because you almost always construct the chord with ti, not te. But that's probably way too much information.
Of course there are more modes or church modes to consider than Ionian mode. There are several ways of explaining the construction of modes, but given the information already contained in this writeup, the easiest way is this:
Take your arbitrary tonic and build a Major Scale in Ionian Mode in Solfege.
do re mi fa sol la ti do
Now move your tonic (your starting point) to another point on the Solfege scale and sing the scale from (new) end to (new) end keeping the same intervals as though you were singing it from do.
For example, to sing:
re mi fa sol la ti do re
in the same key, would be to sing a scale in a different key in the Dorian Mode. Specifically, you'd be singing a scale in Dorian mode in the key of whatever the note value of "re" is from your original Ionian scale. I.E: if your Ionian scale is constructed in E Major, your note value for "do" is E Natural, the tonic and identifier of the key. Your note value for "re" is F sharp. Singing the above example of "re mi fa...re" from E Major in Ionian mode is the same as singing the scale of F#'s Dorian Mode (Which is not a "major scale" by the way - "Major" keys and "Minor" keys are concepts exclusive to Ionian mode (I think)).
mi fa sol la ti do re mi
is theoretically in the Phrygian Mode
fa sol la ti do re mi fa
is theoretically in the Lydian Mode
sol la ti do re mi fa sol
is theoretically in the Mixolydian Mode
la ti do re mi fa sol la
is theoretically in the Aeolian Mode
ti do re mi fa sol la ti
is theoretically in the Locrian Mode
The Locrian Mode is arbitrary, and only exists because of music theory. No plainsong has ever been written using the mode, and nobody in their right mind would compose a piece in Locrian, except for the novelty of it. As a matter of fact, you'll rarely come across many alternate modes in any compositions since the Renaissance era. Every once in a while you'll find a piece in Dorian or Mixolydian mode, (essentially blues scales) and Aeolian mode is/was a trendy way to compose New Age music. Lydian mode is very cool and interesting, but hard to get away with, unless you're Danny Elfman, apparently.
There are many cute little reverse acronyms for remembering the names and orders of modes as well. My personal favorite and easiest way of remembering is "I Don't Particularly Like Modes A Lot," but a more popular one I've found is "I Don't Play Licks My Auntie Likes." Whatever works.
I'd go into the theory of chord qualities and chord progressions based on the Solfege members of the basic diatonic collection, but this already feels like quite a bit of information crammed into such a happy, innocent little node. So Here's Where the Story Ends.