If you're looking for a list of the software options out here, this is a fairly exhaustive list: https://www.capterra.com/task-management-software/
Personally I've used Jira, Trello, and Asana:
Jira: The established leader for more technical task management and is used by organizations big and small for agile software development. I would use this software if your work is technical in nature and you are mainly going to have developers, product managers, and/or designers touching these tasks. In my experience, Jira isn't as easy to use for "non-engineering" roles such as marketing, sales etc. and you could have low adoption trying to push this software to those groups.
Trello: Similar to Jira in its usage mainly for development work and not cross-discipline, but this software is a little lighter weight. This is great for smaller companies who have fairly simple projects they want to track across the engineering/product teams.
Asana: This software is a little simpler to understand for non-tech roles. I've used this well to help coordinate projects cross-division (projects that span multiple teams including non-technical teams like marketing, sales, customer support etc.). Great for keeping the team focused across discipline on each task necessary for larger projects.
Hope this is helpful.
There are many Task management software in the market, some very targeted towards the task delivery approach (Waterfall, Agile etc)
MS Project, Clarity, Clarizen, HP PPM are typical waterfall based software
Rally, JIRA, VersionOne, Asana, Smartsheets, etc for Agile
Question for you : Is it just the Task Management or are you looking for everything related to tasks (duration ,effort, resources etc). Based on the requirement exact software can be recommended
There are several mind mapping and innovation management tools as well like MindJet.
There are many types of task management software for different types of tasks and projects.
Personally, I love managing software projects in storyboards, where I can plan and see the status of everything at a glance across multiple versions. I call it my mission control system.
After a lot of research, I chose StoriesOnBoard.com, which I then sync with a Kanban board on Jira for developers to move tasks through a sequence.
Jira is ultra complex though. It also works just as well with Trello.