Keitra, Click http://www.m-w.com/cgi-bin/dictionary?va=twi-night
and a link will take you to the website for the (American) Merriam-Webster dictionary. There, you will see the definition of the adjective form, 'twi-night'. A 'twi-nighter is just the noun version...ie the name given to such a baseball-match.
I imagine a day-nighter in baseball is the same as in cricket: agame that starts in daylight and finishes at night, under the floodlights. This happens a lot in baseball as many games are played 'after work' during the week, and can drag on (in the nicest possible sense) for many hours...