Jumpgates allow a ship to travel between Sectors. To use a jumpgate click on the 'Use jumpgate' button which will appear once your ship gets close enough to the jumpgate. Note that usually jumpgates cannot be used once you have been engaged in PvP combat. Once you have used a jumpgate you are immune from either giving or taking damage for a few seconds when you appear in the sector on the other side.
Often Mystery boxes appear close to jumpgates.