This is an user contributed article.

Project management software is not just for managing software based project. It can be used for variety of other tasks too. The web-based software must provide tools for planning, organizing and managing resources to achieve project goals and objectives. A web-based project management software can be accessed through an intranet or WAN / LAN using a web browser. You don't have to install any other software on the system. The software can be easy of use with access control features (multi-user). I use project management software for all of our projects (for e.g. building a new cluster farm) for issue / bug-tracking, calender, gantt charts, email notification and much more.

Obviously I'm not the only user, the following open source software is used by some of the biggest research organizations and companies world wild. For example, NASA's Jet Propulsion Laboratory uses track software or open source project such as lighttpd / phpbb use redmine software to keep track of their projects.

You use the following top 10 software for personal or business use. Keep track of all your projects in one place and finish them successfully on time.
#1: Codendi

Codendi is an open-source collaborative development platform offered by Xerox. From only one interface, it gathers, all the needed tools for software development teams: management and versioning of code, bugs, requirements, documents, reporting, tests etc. It is mainly used for managing software project processes.

* Download Codendi

#2: Redmine

Redmine is a flexible project management web application. Written using Ruby on Rails framework, it is cross-platform and cross-database. It includes calendar and gantt charts to aid visual representation of projects and their deadlines.

* Download redmine

#3: ProjectPier

ProjectPier is a Free, Open-Source, self-hosted PHP application for managing tasks, projects and teams through an intuitive web interface. ProjectPier will help your organization communicate, collaborate and get things done Its function is similar to commercial groupware/project management products, but allows the freedom and scalability of self-hosting.

* Download ProjectPier

#4: Trac

Trac is an open source, web-based project management and bug-tracking tool. Trac allows hyperlinking information between a computer bug database, revision control and wiki content. It also serves as a web interface to a version control system like Subversion, Git, Mercurial, Bazaar and Darcs.

* Download Trac

#5: Project HQ

Project HQ is a collaborative open source project management tool, similar to Basecamp and activeCollab. Project HQ is built on open source technologies like Python, Pylons and SQLAlchemy and is fully database independent. Project HQ uses a structured workflow to assist you in managing your projects.

* Download Project HQ

#6: Collabtive

Collabtive is a web-based project management software that is being published as Open Source software. The project was started in November 2007. It strives to provide an Open Source alternative to proprietary tools like Basecamp or ActiveCollab.

* Download Collabtive

#7: eGroupWare

eGroupWare is a free open source groupware software intended for businesses from small to enterprises. Its primary functions allow users to manage contacts, appointments, projects and to-do lists.

It is used either via its native web-interface, making access platform-independent, or by using different supported groupware clients, such as Kontact, Novell Evolution, or Microsoft Outlook. It can also be used by mobile phone or PDA via SyncML.

* Download eGroupWare

#8: KForge

KForge is an open-source (GPL) system for managing software and knowledge projects. It re-uses existing best-of-breed tools such as a versioned storage (subversion), a tracker (trac), and wiki (trac or moinmoin), integrating them with the system’s own facilities (projects, users, permissions etc). KForge also provides a complete web interface for project administration as well a fully-developed plugin system so that new services and features can be easily added.

* Download KForge

#9: OpenGoo

It is a complete online solution focused on improving productivity, collaboration, communication and management of your teams. OpenGoo main features include document management, contact management, e-mail, project management, and time management. Text documents and presentations can be created and edited online. Files can be uploaded, organized and shared, independent of file formats.

* Download OpenGoo

#10: ClockingIT

ClockingIT is a free Project Management solution, which helps your team stay focused and on top of things.

* Download ClockingIT

Ed: The following two paragraphs added by Vivek Gite:

I also use project management software to keep track of how much time I spent per client and project.
My Personal Choice

redmine is my personal choice because I like to use ruby on rails and I often work with small teams. We track networking issues, data center issues, capacity planning, trouble tickets and much more using redmine. I can track multiple projects and its flexible role-based access control make sure only authorized eyes can view the details.
Other FOSS Project Management Software Projects

1. JotBug
2. Bugzilla (only bug tracking)
3. OpenProj (desktop app - replacement for MS-project)

http://www.cyberciti.biz/tips/open-source-project-management-software.html