Productivity and Task Management

There is a huge market for productivity and task management software, and an accordingly large number of applications and subcategories to fill the niche. This includes to-do lists, calendaring and hybrid forms for both mobile and non-mobile platforms.
There are multiple pages of apps filed under productivity in both iTunes and the Google Play store. Many can be used on your phone, as well as online from any computer.

Here are some that are good and FREE:
Remember the Milk (prioritize assignments, estimate time, add tags, due dates)

Google Calendar (mostly for scheduling, easy color-coding, can add notes to events)
Soshiku (manage high school assignments, set-up alerts via email or text)

Toodledo (Can integrate with online calendars, collaborate and share tasks with others, very customizable)

Brainstorming

There are a number of different apps for smart phones or computers that will allow you to visually brainstorm (and some let you share and collaborate with others). This type of tool would be useful during the beginning stages of research projects or essay outlines.

Some available programs/apps to check out:
XMind
FreeMind
Mindomo
Mindmeister
Cmap
Comapping (real-time mind map editing from multiple users; ideal for group projects)

Here's a sample screenshot from FreeMind: