Mature means the wine has evolved from its youthful classic style into a just-as-interesting aged style. More flavours called tertiary notes appear. This can be after 5 years, or after 10, or 50, depending on the wine.
If you've not "done it" before, start with Riesling, Petite Arvine, Gamay, Syrah or Chardonnay as they age in a most accessible manner.
Graacher Domprobst Riesling Beerenauslese Weingut Phillips-Eckstein
Bernkastel-Kueser Kardinalsberg Riesling Beerenauslese Weingut Anton ZimmermannBeerenauslese Riesling keeps forever.