Solfège is a system for singing notes. If you're familiar with the famous Rogers and Hammerstein song "Do-Re-Mi" from The Sound of Music, you already know the solfège note names: do, re, mi, fa, sol, la and ti.
The first and last syllables have variants which are a matter of custom: In France, ut is sometimes used in place of do. (ut is rare, so we'll ignore it here.) Similarly, si is used in continental Europe while ti has been used in English-speaking countries since the 19th century. (The reason for changing si to ti was so that each syllable would begin with a different letter.)
The original solfège—also called solfeggio and solmization—note names derive from an 11th-century hymn by Guido d'Arezzo, in which the solfège syllable is the first note of each phrase. The starting notes of each phrase are C, D, E, F, G, A:
Latin-derived languages (Italian, French, Spanish) assumed these names, and ut was eventually changed to do in Italy and later in the other countries. The solfège syllables are thus nothing more than the note names in French, Italian and Spanish, minus any accidentals (sharps or flats).
Here is a pronunciation key for the primary solfège syllables:
Because these are the note names in these languages, in these countries fixed do solfège is taught, which means that the note names are sung regardless of key. Thus, C, C-sharp, C-flat, C double sharp or double flat are always do, no matter their harmonic context. Similarly, D, D-sharp, D-flat, D double sharp or double flat are always re, regardless of what key the music may be in.
The solfège syllables are not necessarily the note names, however. There is an alternative system called movable do, in which do is always the tonic. For example, in C major, C is do; in D major, D is do; in E-flat minor, E-flat is do, and so on. The remaining syllables continue up each scale.
Here is C major in movable do:
... and D major in movable do:
Notice how the solfège syllables remain the same when going from one key to the next?
Movable do contains additional syllables. Minor keys, for example, have different syllables, because certain notes are lowered with reference to their parallel major scale. Here are the three variants of the minor scale, with movable do solfège syllables:
A few examples of solfège scales will suffice to illustrate the difference between fixed and movable do:
C major in fixed do:
C major in movable do:
C harmonic minor in fixed do:
C harmonic minor in movable do:
D major in fixed do:
D major in movable do:
D harmonic minor in fixed do:
D harmonic minor in movable do:
In all of these examples, you can easily see the difference between the two forms of solfège. Movable do may be suitable for beginning- to intermediate-level solfège ear training, but it quickly gets complex and confusing and the system in fact breaks down. Music is by no means always well-behaved! Harmonic sequences for instance will often cycle rapidly through keys without being firmly in any particular key. Movable do will either fail or become exceedingly confusing in such cases, defeating the purpose of solmization. Fixed do, by contrast, works very easily in all music, tonal or atonal, regardless of harmony or lack thereof.
This is the reason why I teach fixed do solfège on key-notes. It's ultimately far simpler. If you see any kind of C—C natural, C-sharp, C-flat, C double sharp or C double flat—sing do. If you see D of any sort, sing re, and so on.
Fixed do is not without its own disadvantages, however. The syllables do not transpose to other keys, and intervals between the same scale degrees are sung using different syllables, unlike movable do.
In the end, however, I highly recommend using the fixed do system of solfège for its simplicity, as it will become more and more useful as the music you learn becomes more advanced, whereas movable do becomes less useful as music increases in complexity.