If a dotted half note precedes a quarter note of the same pitch, is the note held for the duration of the former or latter? Does the following quarter note supersede the dotted half in this case?
That dotted half note is tracked to a different voice; it basically is like another melodic line played at the same time as the other voice (the way you can tell is if the stems of the noteheads point away from the center of the staff). The dotted half is played for the entire measure, whereas the other notes are all played in sequence starting at the same time that the dotted half does.
In the future, for questions like these, it is invaluable to post the time signature of the piece (which I assume to be 3/4).