The Piano Trill
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This piano ornament may be written in various forms:
All of these signs above mean exactly the same.
(In the Baroque style it is written slightly different:
This sign 
is confusing because it is in the Baroque genre, but in other genres it means
an upper mordent.)
It's a rapid alternation between the written note (the principal note) and the note above it. In the example above the principal note is G, so it contains two notes: G and A.
It always begins in the upper note and ends in the lower one.
How many notes should be played in this piano ornament?
It depends on the tempo of the piece, the style, the skill of the performer and his taste. The minimum number of notes in this piano ornament is four.
Here are some examples:
Four notes in various rhythmic interpretations:
Six notes in various rhythmic interpretations:
An eight notes trill:
As I said before, the way it should be performed is not constant. You can perform it continuously through the whole value of the principal note, or you may play quickly just a few notes and then rest on the principal note.
A trill with termination (a turn)
It ends in some cases in a turn. This means that at the end of the trill you should play a note that is one step below the principal note instead of the note above.
The following example:
Should be performed this way:
One of the most famous examples of a turn is from the popular Sonata K 545 by Mozart:
Here is how the measure should be played:
The audio example is rather slow in order to help you understand how it should be played
Now, I would like to add some important comments that I advise you to read.
As a piano teacher, I recommend to play the whole piece without referring to piano ornaments at first. It means that when you see a note with a trill above, it is better if you ignore it. Only when your playing is fluent should you try to perform the ornaments.
One more comment: Some editors write above the exact notes that should be played. You may find this very helpful.
