I am wondering how the accidental in the first chord (see what is circled) is played? Does any accidental simply move the note up or down a half-step from what the note is supposed to be based on the key signature or does it override the key signature all together? Specifically, is the accidental G# played as a natural G (since G is already a flat in the key signature - thus a half step up) or actually as a G#?
The chord played is Amaj7, made up with A C# E and G#.The key sig. is Db/Bbm.I guess that the G# is shown instead of a possible Ab, which is technically correct for that chord.
All accidentals over-ride the key sig., for the rest of the bar they're marked in.Sometimes the author will be helpful and remind the player that a particular accidental is not needed in the next bar by using another accidental,which can get confusing !
Any accidental will show the actual note to be played, not just sharpening a flat, for example. In that key sig.,the Gb is changed to a G# with an accidental. It doesn't 'sharpen the flat' to make it a G natural.
Accidentals override key signatures and previous accidentals. The circled chord has two G# and and one C# note. Having "additive" accidentals would make it very hard to read music. In this excerpt, the next octave chords in the top staff would then be B-flat, then B-doubleflat, and then either G natural (if adding to the previous accidental) or or G double flat (if adding to the key signature)...
As has been mentioned in other answers, alterations are not relative, they're absolute. So a sharp makes the note sharp, regardless of the alteration it had due to the key you're playing in; so if the key states Ab and you suddenly see an accidental sharp on an A, you play A#. Same goes for the inverse case.
So I just wanted to add that there is a sign to remove an alteration: the natural. Your example includes the natural sign in the second chord on the treble clef; that one is used to "neutralize" a sharp or flat, so you play B natural there (which is what you were thinking the sharps did in the first chord).
The key signature is always secondary to any accidentals next to the note. In the chord you circled you would play a G#, C#, and another G# instead of a Gb, C, and another Gb. They never cancel each other out just play what is written.
Accidentals take precedence to any alteration from the signature. Moreover, they apply until the end of the bar for every pitch. If there is a G# in a bar, then, until a new accidental or the end of the bar, all the following Gs on the same octave should be played sharp.
The key signature is not overridden. This is just a chord borrowed from a different key and the accidentals continue until the end of the measure unless otherwise noted.
As noted in the other replies, accidentals are "absolute", with the meaning unfazed by preceding material or key signature. There is one exception: when "weakening" a previous double accidental (key signature or previous in the bar), the resulting single accidental is often printed combined with an immediately preceding natural sign.
While the existing answers are correct for modern music, there was a time before the invention of the natural sign in the middle of the seventeenth century, when music was less chromatic and harmonies rarely strayed very far from the tonic. In those days, the sharp sign was used to cancel the flat sign.
An example may be found at the end of the first Kyrie of Byrd's Mass for 4 voices, the last note of the second staff in the image.
