Beyond simple task management is project management, which requires a much more sophisticated approach to tasks and schedules. One app to consider on the iPad for such work is the $20 OmniFocus from the Omni Group. It syncs with the desktop (Mac-only) version of the software and ensures the Calendar app has your due dates added. It even can show nearby task locations on a map, such as for running errands or scouting film locations. Another well-regarded task manager is Cultured Code's $20 Things for iPad, which boasts a clean but capable user interface, a "smart" today list, and scheduling features.
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For a more personal management tool, take a look at FileMaker's $5 Bento for iPad, which is a lightweight database perfect for tracking records, contacts, to-dos, and other personal tasks and data.
For more work-oriented mobile databases, an app worth considering is DDH Software's $10 HanDBase for iPad.
Two other noteworthy apps give you remote access to full-fledged database management systems from the iPad. The $40 FileMaker Go for iPad  lets you query and edit FileMaker Pro databases, while the $2 SQLTouch lets you run queries and manage records in MySQL databases.
Mind-mapping and whiteboarding
If you do mind-mapping -- a method of organizing thoughts and items by their relationships -- CMS's $10 iThoughts HD could be the tool for you. It works with several mind-map formats and lets you export your mind maps to PNG and PDF. It also syncs to Box.net and Dropbox cloud storage services.
Markus Müller's $6 MindNode is another mind-mapping tool to consider. Among its capabilities is support for VGA output to a monitor, provided you have a VGA adapter for your iPad.
If you're looking for a tool to do either mind-mapping or traditional whiteboarding, Pixelglow Software's $10 Instaviz is a strong option.
For more traditional whiteboard sketching, give Avici Software's $5 Whiteboard HD a look. It gives you controls over labeling, provides a grid background to help in placement, and supports image inclusion from the Photos app. It also works with an iPad VGA adapter for presentation on a full-size monitor.
Diagramming and charting
For sophisticated diagramming of process charts, website wireframes, or other complex relationships, the Omni Group's $50 OmniGraffle is highly regarded. It supports freehand drawing but also has a bevy of intelligent object formatting and page layout controls. Its files are compatible with the desktop version, and you can export PDF view-only versions as well.