Some of the ways quests can be started:-
Quest nearby will show some of the availble quests in the areas. Note it doesn't show every quest and there maybe more that can be started by talking to the npcs in the area.
Following multiple posts from several players on board 8 suggesting that event quests be excluded from the standard deeds leaderboard, in March 2015 quests were split into three types deeds, feats and legacy. Each type was given it's own leaderboard and listing/profile commands.
These are usually multiple stage quests which can be accepted by talking to a questor or by typing quest accept <no> where available.
Single stage achievement quests these are sometimes awarded when killing a particularly difficult or legendary monster.
Quests that are no longer available and have since closed for completation. These are usually limited edition quests given out during a event.
Apart from the satisfactions of completing them, quests may also offer a variety of different rewards depending on their type and level:-
The type and level of the award is determined by the builder, some of the best equipment in the game is given out by completing quests. The gold given on individual quests tends to dent the more characters that complete it.
It is not possible to remove or restart quests once started without completing them and there are no current plans to add this feature, as doing so would fundamentally undermine the current structure of quests.