Pharmacy students are not required to do a residency, but ASHP has stated that their goal is to push for all future PharmD students to have one year of residency by 2020.
Granted, it most likely won't apply to many of us here yet. Currently, it is optional to do a residency. There are a variety of residencies to complete; some are available through hospitals, and others can even be available in the community setting. It is basically a way for someone to gain more exposure and receive much more education, while being a practicing pharmacist. People are paid lower than a full-time pharmacist during this time, but are receiving a lot of education during this time.
Depending on the residency, generally people will "rotate" through different sites in a hospital residency.. such as Ambulatory Care, Pediatrics, Oncology, etc. This will give pharmacy residents the opportunity to explore various specialties and to find "the place" for them. A second year residency (PGY-2, PGY-1 = First Year) will consist of a specialized concentration in one department generally, such as oncology.
In the past, hospital pharmacy positions did not require residencies. However, in the Chicagoland area, jobs are becoming much more competitive, and having residency experience can definitely help. (Depending on what you want to go into). It might not be as necessary for community pharmacists, but I do know that Dominick's (Safeway in Illinois) has recently hired many people with residency experience.
Most people don't decide until their 3rd or 4th year whether residency is really right for them. Especially with pharmacy school, a lot of people are pooped. Last year, 1/3 of applicants did not get matched for a residency.
TL;DR: Residency follows pharmacy school* and is a nice way to gain more exposure and actually practice with a pharmacy license, while getting much more education and exposure in a pharmacy setting of your choice. There are 1 year and 2 year residencies and they have the potential of boosting your resume and experience, but are not required.
* - You don't have to go right into it; you can even work for a while after pharmacy school, and apply later for a residency.