Sprints are the way that Scrum teams iterate over software. The goal of a sprint is to produce a working set of software that is potentially shippable. I say potentially because it's really up to the business to decide when the software is shippable. As long as the feature development is complete there is a choice. You can evaluate the number and severity of open bugs as well as the other features and make a call on whether you ship or not.
Sprint teams are comprised (at minimum) of a representative of product and the software development engineers. Though in practice you'll really want someone from QA (quality assurance, if your org has a separate QA group) and someone from operations represented.
In Agile we favor collaboration over contract negotiation. The role of product management as part of the scrum team is to represent the voice of the customer. The product manager should be working directly with the customer to understand their needs. The product manager helps collaborate with the rest of the team on the requirements of the feature(s) being developed each sprint.
The developers are the ones actually building out the features. The role of the developer is to collaborate with product to turn the requirements into feature stories and tasks. The developer is also the advocate for the health of the code and architecture. The developer is responsible for making sure that unnecessary tech debt isn't accrued sprint over sprint. Often this means collaborating with product on the scope of work for each sprint to account for taking on bug fixes and technical debt.
Quality Assurances Role
Ideally all software we write includes unit, functional and integration tests. But that doesn't mean that all testing can be automated. There are some aspects of the user experience that may require manual testing. Or if your organization is new to agile you may have creating automated tests as tech debt. Whatever the reason, our goal of building potentially shippable software each sprint means including QA as part of the sprint team.
One other often overlooked member of the sprint team is operations. While continuous delivery and continuous integration are the ideals for deployment, many organizations have not shifted to that model yet. If your org is one that hasn't, including operations will help accurately account for the sprint work necessary to release the software.
Each sprint has a scrum master. It is the role of the scrum master to facilitate scrum. This is done by managing the product backlog, facilitating the scrum meetings, facilitating the sprint planning meeting, facilitating the retrospective meeting, helping to define what done means for a given story or task, and removing roadblocks to the team. The scrum master is also responsible for removing roadblocks to the scrum team. This is usually done by fostering collaboration between teams and product.