This java applet demonstrates various properties of vector fields. You can select from a number of vector fields and see how particles move if it is treated as either a velocity or a force field. This helps you visualize the field.
The menu in the upper right has a variety of different fields to choose from. You can also input your own by scrolling to the bottom of that menu and selecting "user-defined field" or "user-defined potential".
See also the 2-D version.
Zip archive of this applet.
Version 1.3b, posted 5/27/11