A big distinction to be made is the specialties where the chief resident has already graduated residency and is functioning as an attending (e.g. IM/peds) and the specialties were the chief resident is just a senior resident (e.g. surgery/surgical subspec/radiology/etc..). If the former situation, the person is doing an extra year in their training/academic paradigm. In the latter, the person is not.
Why do people become chief residents?
What do they do (compared to other residents)?
And do they typically pursue fellowships afterwards?
Why become a chief? Lots of different reasons to become a chief. Vox hit on some of the more career-enhancing aspects of it. His points are valid, though not universally applicable. Chief residency is considered an honorific of sorts. If chosen, you were deemed to have the leadership and administrative abilities above and beyond your colleagues. It's a very desirable line on a CV for any job (academics or PP).
I personally took on chief residency for different reasons than Vox mentions. I was in a medium sized radiology program and the program essentially let us self-select our 2 chiefs. Of 8 people in my class, half had no interest in the position. Of the other 4 (including myself), I thought two would do a terrible job. The chief resident has a lot of autonomy over the daily schedule and it's ripe for abuse with the wrong people. The other person I liked and I banded together and convinced the rest of the class and the PD that we were the best choices and they agreed.
(Side note: the class after mine chose people who scheduled a ton of academic days for themselves and their cronies and it was a bad look for everyone).
What do chiefs do? variable among programs and specialties but some of the bigger things are
Scheduling: We made the yearly rotation schedule and were primary managers of the daily schedule. (i.e. someone called out sick, we dealt with that).
Education: not my program in particular but in some programs the chiefs are responsible for organizing the resident lecture series
Interviewing: My co-chief and I participated in every residency interview day. We collectively interviewed every single candidate to the program and subsequently participated in rank order list discussions.
Then there's the people skills/putting-out-fires portion:
Being the go-between: Addressing issues that arise between the residents, the PD, your department's faculty, the hospital (/hospital leadership), and other departments. This can be a massive headache.
Being the bad-cop: Sorta related to the above point. Everyone's least favorite part - dealing with problematic/struggling residents. Not uncommonly did I have an attending grab me and tell me I needed to have words with a resident about their performance, attendance, or even attire (seriously). I did pull residents aside to have brief chats about those concerns, hopefully in a non-threatening way. If things didn't improve, I had to elevate those concerns to the PD. In our monthly residents meeting that my co-chief and I lead, we often had to tell people to do basic **** like show up on-time, show up to lecture, or log their duty hours.
Being the shoulder to cry on: The chief is commonly the first person people go to with personal problems (that may become work problems). I know it from experience, cuz I hassled my chiefs with my own stuff. Some of the stuff we dealt with: resident going through divorce, resident going through a serious healthcare issue, resident under CPS investigation, resident going through a tough pregnancy, resident struggling with single-parenting. Just whoosh.
For me, all that was on top of just being a senior resident. Depending on the program/specialty, "chief" responsibilities can be hoisted on 1-5 people.
Do chiefs pursue fellowship afterwards? Depends on the individual, specialty and goals. One of my ex's did an IM chief year and then went into primary care; her co-chiefs all did fellowship afterwards. In radiology, everyone does fellowship anyway.
How does one become a chief resident? What criteria? Process?
This varies from institution to institution. Some places it's more a decision of the PD or the faculty. Mine was more resident self-selected.
In general, you just pick your best residents. The ones that are the most clinically competent and have the best people skills. Some places put a lot of weight on the academic side; i.e. choosing from the high ITE and scholarly achievers.
The people picked are the ones that are predicted to best handle the load of chief responsibilities *on top* of their clinical work.