Solfège is a system of naming musical notes with syllables to indicate their position in a key. It was first used in about the 10th Century to help educate monks who needed to sing, but couldn’t read. This has evolved into about 3 different ways to use similar syllables today.
The syllables are do (doh), re (ray), mi (mee), fa (fah), sol (sole), la (lah), and ti (tea). I usually write so for sol, because the “l” is unhelpful in singing warm-ups. These syllables line up with the notes of a major scale.
You can also think of “Do, a deer” from Sound of Music: https://www.youtube.com/watch?v=1RW3nDRmu6k
Some places used fixed-do solfège, where do is always the note C. This type of solfège is useful if you’d like to have perfect pitch, and is used pretty widely in European (French) conservatory lineages. There are definitely reasons to use fixed-do, but is less common in the US.
If you are in the US, you are more likely to come across moveable-do solfège, where do is always the tonic or name of the key. There are two types of moveable-do: do-based minor and la-based minor. In do-based minor, more syllables are added for sharps and flats, do continues to be the tonic in minor: do re me fa so le te do. (me, le, te are all lowered a half step from their versions in the major scale.)
In la-based minor, the tonic shifts to la, so that a minor scale would be la ti do re mi fa so la.
Many choral traditions use la-based minor, and it is good for music that is ambiguous about whether it’s in the relative major or minor. On the other hand, do-based minor is very common in university music departments, and is good for music that moves between the parallel major and minor, as well as identifying functions of notes and chords. Do-based minor is my go-to solfège, so that is the one I tend to use (unless otherwise noted) throughout this site.