If you don't want to do the rest of a quest you can usually go back to the person you got it from and there is an "I don't want to do this quest right now" option. Not necessarily worded like that though. Also if you talk to a new NPC they might say "You are already working for someone, do you want to work for me instead?" and they will cancel the previous quest and put you on their's.
There are different kinds of quests of coarse, meaning ones that show up in your datapad and ones that go in your quest log. Even some that don't seem to show up in either place.
The ones in the Datapad can be dropped simply by choosing to abort it from the radial menu. Quest Log quests can't be dropped from the log, and seem to be permanent until completed. NPC quests can be dropped by talking to them and choosing to quit, which may work for some quest log quests also.