Performance Testing Tools:

Here are some important tools to test the performance, load and stress of the website/application. These tools ensure that your website/application will run under high performance and less load & stress.
Apache JMeter is a 100% pure Java desktop application designed to load test functional behavior and measure performance. It was originally designed for web applications testing but has expanded to other test functions since.
LoadRunner is a load testing tool for Windows and Linux, given by HP to test the web and other applications very efficiently before it goes to end users. It is available in stable version 12.0 in many languages. LoadRunner is very helpful in determining the performance and result of the web application under heavy load (load due to thousands of users accessing the web application at a same time).

Automation / Functional testing Tools

Most of the bugs come due to functional issue of the website/application. So, here’s some web automation/functional testing tools.
Selenium is one of the most popular automated testing suite. Selenium is designed in a way to support and encourage automation testing of functional aspects of web based applications and a wide range of browsers and platforms. Selenium suite is comprised of 4 basic components; Selenium IDE, Selenium RC, WebDriver, Selenium Grid. Selenium IDE is Firefox add-on for record-and-playback web application tests. WebDriver directly communicates with the web browser and uses its native compatibility to automate.
Thucydides is a Java-based open source library designed to help teams implement Acceptance Test Driven Development more effectively. Thucydides was a Greek historian known for his astute analysis skills who rigorously recorded events that he witnessed and participated in himself. In the same way, the Thucydides framework observes and analyzes your acceptance tests, and records a detailed account of their execution.

Testing and Project Management Tools:

Test management tools are used to store information on how testing is to be done, plan testing activities and report the status of quality assurance activities.
TestRail is a centralized test case management tool – you can use it to create test cases and test suites, track execution and report metrics. Additionally, it integrates with many issue tracking tools that makes requirements from external systems to be linked to test cases in TestRail; bugs can also be created in the external systems and links can be established to the corresponding test case. It comes with a HTTP-based API to integrate with the automated test results. One most common integration is with Gemini, which is a incident/ticket management system(supports agile too).
TestLink is one of the very few open source test management tools available for use in the market.  It is a web-based tool with typical features like, requirement management, test case creation and maintenance, test runs, tracking bugs, reports, integration with common issue trackers etc.
Confluence is team collaboration software. Written in Java and mainly used in corporate environments, it is developed and marketed by Atlassian. Confluence is sold as either on-premises software or as a hosted solution.

Bug Tracking Tools:

All IT projects needs a bug tracking (or issue tracking, or defect tracking) system.
JIRA is a proprietary issue tracking product, developed by Atlassian. It provides bug tracking, issue tracking, and project management functions. Although normally styled JIRA, the product name is not an acronym, but a truncation of Gojira, the Japanese name for Godzilla. According to Atlassian, JIRA is used for issue tracking and project management by over 25,000 customers in 122 countries around the globe.
MantisBT is an open source issue tracker that provides a delicate balance between simplicity and power. Users are able to get started in minutes and start managing their projects while collaborating with their teammates and clients effectively. Mantis Bug Tracker is a web-based bug tracking system released under the terms of the GNU General Public License version 2. The most common use of MantisBT is to track software defects. However, MantisBT is often configured by users to serve as a more generic issue tracking system and project management tool. The name Mantis and the logo of the project refer to the Mantidae family of insects, known for the tracking of and feeding on other insects, colloquially referred to as “bugs”.
Redmine is an open source issue tracking system that integrates with SCM (source code management systems). Even though it is not a ‘bug tracking’ tool it involves working with issues, where issues can be features, tasks, bugs/defects etc. Redmine is written using the Ruby on Rails framework. It is cross-platform and cross-database and supports 34 languages.
Bugzilla is a web-based general-purpose bugtracker and testing tool originally developed and used by the Mozilla project, and licensed under the Mozilla Public License. Released as open-source software by Netscape Communications in 1998, it has been adopted by a variety of organizations for use as a bug tracking system for both free and open-source software and proprietary projects and products. Bugzilla is used, among others, by the Mozilla Foundation, WebKit, the Linux kernel, FreeBSD, GNOME,KDE, Apache, Red Hat, Eclipse and LibreOffice.
Pivotal Tracker is Pivotal Labs’ software as a service product for agile project management and collaboration. In July 2011,Pivotal Tracker had over 250,000 registered users. The tool includes file sharing and task management, velocity tracking and iteration planning; release markers; and progress charts. There is an API for extensions and third party tools.
 
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References:

• 30+ Most Popular Web Application Testing Tools – Comprehensive List with Download Links at softwaretestinghelp.com
• 15 Best Test Management Tools for Software Testers at softwaretestinghelp.com
• Top 10 Open Source Bug Tracking System at thegeekstuff.com
• Pivotal Tracker | Agile Project Management at www.pivotaltracker.com
• Mantis Bug Tracker at www.mantisbt.org
• Wikipedia