In Hitman 2 there are two types of objectives. There are the mission objectives, which are generally just the overall requirements to complete the level. And then there are side objectives or challenges that can earn you Mastery Points and Weapon Unlocks.
You can keep track of your challenges and see which ones are available at any time during a mission. Simply press the touchpad (PS4) or the View “back” button (Xbox One). Once in this menu you can look at your Assassination Challenges, Discovery Challenges, Feats, Targets, and Classic Challenges. In this menu you can further drill down into the Challenges that you’ve completed and which ones are available to you by toggling the view by pressing the Triangle/Y button.
Completing any of these challenges will earn you extra Mastery Points which can unlock new weapons, tools, and other items to bring with you on future missions. Some challenges will reward you straight away with a weapon or item unlock as well.