As the Station Psychologist (Or a Psychiatrist or a Therapist), you are tasked with identifying and solving personal and mental issues within the station's crew. If need be, you have the power to deem someone mentally unstable, and ask their supervisor to demote them.
You are not another Doctor, your job is to help mentally, not physically. This is largely a roleplay-based role - whether you have fun with it is largely tied to how much effort you and your patients put into it, as it is not tied to game mechanics at all.
Psychology and You
You are now in charge of the mental well-being of the entire station! Nanotrasen has entrusted you with the task of keeping everyone around you stable or contain them, if they are beyond your help. It is worth noting that much of the crew would deem themselves to be perfectly sane, and it is up to you to determine the psychological status of those around you (especially those in command roles).
First, it is a good idea to start 'advertising' around the station for anybody who could use some psychological care, and make sure to check the Brig, Medbay, or even places such as the Bar for any potential patient that could need some help. Checking medical records on your office's laptop can be a valuable tool, aswell - people who bother to write them are much more likely to be receptive to roleplay - do not be afraid to shoot them a PDA message explaining the situation! Being the nice, helpful person everyone wants to meet in their life can definitely help. Lastly, keeping track of the current events unfolding on station can be useful, too - who knows when an Assistant will need help with the Post Traumatic Stress Disorder they acquired after seeing the Chef brutally murder and gib Pun Pun.
After doing this, and after you've got a lost soul that needs your help, it's time to move to the next step!
You have found someone in need of help, and managed to convince them to come to your office. Now, the fun begins! There is no strict, "correct" way of treating people, as it is all roleplay-based. This means that you and your personal ideas are in the spotlight! Talking to your patient is always a great idea, and many great RP moments come from doing so - ask them about their problems, their feelings, perhaps even take notes and write a report on what you have found.
After ascertaining the issues plaguing the poor crewmember, you may commence treatment. Perhaps their issue is wholly possible to solve with just psychotherapy, and perhaps, they need medication - communicate with the Chemist or Botanist if you plan to use medication extensively, as the medicine you start out with can run out rather quickly. A quick look at the Guide to Chemistry might give you some ideas about what experimental drugs to use. Paperwork helps with making your treatent seem more legitimate, and a formal prescription will save your patient some trouble with requesting the medication. Keep in mind, however, that forcing medication on people can be a crime, and hurting people as a non-antagonist might even break server rules.
Should you deem someone unfit for their duties, and beyond treatment, it is in your power to suggest their demotion. Simply contact their supervisor, or in the case of Heads of Staff, perhaps the Nanotrasen Representative, and explain the situation. This, of course, should be the last recourse. Having to go that far means that your treatment has failed. Again, paperwork helps - a well-written report is harder to ignore, and will seem much more legitimate than word of mouth.
Security and You
It's possible you might eventually bump heads with Security. After all, many criminals are mentally impaired and many of our less stable friends turn to a life of crime. Now, before you write a report damning the security department, there are a few options to consider. The first thing you should do is ask the Head of Personnel and Head of Security for access. If you have reason to believe you need access, they might grant you brig access to treat those insane convicts. Otherwise, don't mention it unless you REALLY need to get in there. Then go to Internal Affairs and complain. You could also ask the Captain. If they are not busy, they may help you. Remember - matters of Security are ultimately more important than yours, and thus your requests might simply be ignored.
You'll have a hard time trying to gain access somewhere that isn't part of the medical department, but unlike most medical professionals, going around the station a lot will not necessarily arouse suspicions, assuming you can use the excuse of 'Searching for new patients'!
Your office contains no cameras or intercoms, can be accessed by few people, contains a locker with the same, uncommon access as your door, and depending on circumstances, you can bait your target - or someone with a useful ID or equipment - into it. From there, they can be highly vulnerable. Drug or stun them, and then kill, rob, or even mindslave them! You can invite other baddies into your office while pretending to give them group therapy and discuss your nefarious plans, safely outside of the reach of the AI! Use your medical access to steal medicine, and potentially even clone your buddies while the Coroner is not looking! As the Psychologist is a roleplay-based role, your creativity will be the determining factor when it comes to your success, and it can lead to truly interesting moments that everyone involved will remember for a long time.