Scordatura

“Scordatura” - alternate tuning.  It’s not just for guitars.
 
Scordatura is more common in music of the renaissance and baroque periods, when western classical music didn't have centuries of "standard practice" behind it.  But it has been used in later music as well, such as the viola in Mozart’s Sinfonia Concertante (all strings tuned up to make the viola louder), or in Kodaly's Solo Cello Sonata in B minor (the C string tuned down to be able to play a low B).
 
There is no one "standard alternate" tuning.  Scordatura refers to any type of non-standard tuning:  tuning all strings up or down, retuning one or two strings up or down, etc.  Sometimes a string is retuned by a half step, sometimes a whole step, sometimes even more, though there's a limit.  Tune a string too far down, and it becomes too flabby to produce much sound; tune it too high, and it will break.
 
There are a few reasons why a composer might want to use Scordatura.  It might make certain chords possible which can't be played with standard tuning, such as the crass open-stringed tritone of the solo violin in Saint-Saens' Danse Macabre.  It might make an instrument sound softer or louder.  (Retuning each string higher makes them tighter and therefore louder.)  There might be other effects that can be achieved by retuning, such as in that jokester Haydn's Symphony #60, which stops mid-movement while the violins comically retune their G strings up from an F.  Or it may be simply that the composer wants the instrument to play a note lower than its natural range.
 
So, how does it work when you actually have a sheet of music in front of you?  Well, there will be an indication at the top of the score showing what pitches to tune to, like this:

In this example from Bach's Cello Suite #5 in C minor, the top string is tuned down a whole step to a G.  Usually, then, the musician plays as if the instrument is tuned in the standard way.  Then this note, which is 3rd finger on the A string and normally a D:

...is still played with the third finger on the same string, but is now a C.  So for those of you with a good ear, you have to ignore what you're hearing until you get used to it. I found that it helped a lot to learn the suite first in standard tuning so you know how it goes, and then learn the Scordatura version.  In this suite, some of the chords will be different - because you can - but it's pretty easy to figure out what to do.
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Here's a quick recording I made in my studio of the Gigue from the suite.