My understanding is that the chief resident(s) are usually chosen in the second or third year of residency (for a three year residency) by the program director and a group of faculty. The chiefs are usually chosen based on their performance thus far in the residency, leadership and teaching skills, etc. They usually make a list of the people they want to ask in a given year (kind of like a rank order list) and then keep going until they have the number of people they need agree (usually 1 or 2).
The sacrifice is that you spend one that you could be doing a fellowship or enter practice, etc. The advantages are obviously the experience you get from being a role model and mentoring the residents (particularly the interns), honing your teaching skills, learning a lot about the administrative aspects of academic medicine, and of course it is wonderful to have this on your resume.
I know some programs even choose their chiefs while they are still interns. One program does this and requires that they do something else for a year post residency before they come back to be chief for a year , i.e. first year of fellowship, attend on the wards, etc. The process varies everywhere.
Hope this helps.