An ID card is one of the most important pieces of equipment you can have. It allows you to open Airlocks, and be identified by other people, amongst other things.
Your ID card is equipped in the id slot, when you are wearing a Jumpsuit. If you take your jumpsuit off, you will also drop your ID card. Your ID card will publicly display a name and a job. Additionally, the card will have a set of permissions contained within it. These control which doors you can open - you will have to ask another crewmember or the AI to open the door for you if you don't have high enough permissions. APC's and secure lockers require you to swipe your ID over them to unlock them. You need to take your id into your hand to do this.
You can have your job changed and your permissions upgraded by talking to the Captain or the Head of Personnel, who should have access to a machine. You will have to give them your card (by putting it on a table or dropping it) in order for them to change your details.
If you hide your face with a mask, you can appear as someone else by wearing their ID card or appear as 'Unknown' if you're not wearing one. If you show your face however, people will recognize you and notice you are messing with your id. If you speak your voice will give you away too if you're not using a voice changer.