An "Assignment" is just a set of lights paired with one or more commands to be sent to those lights, and these are represented by colored circles in the preset editor, or next to each name in the presets list. You can have as many as you like, though any given light can belong to only one of them. For example, in the Rainbow preset, some lights will be red, some yellow, some blue (if you have that many light bulbs) but obviously no individual light bulb can be more than one color at the same time.
There are currently five different types of Assignments, representing the various classes of commands you might want to send to a lightbulb. To add a new assignment, just tap the gear menu to the right of the existing assignments in a preset:
You'll now see a panel with various things you can do with your preset's assignments, but for now, choose one of the assignment types under "Add New Assignment". Picking "Colors" gives you full control over everything from a single static color to a complex animated loop of colors. "Whites" will present you with a slider for brightness and color temperature, which you can think of as a spectrum of warm and cool whites for everyday lighting. "Brightness Only" will send a command to lights to preserve their current color but only modify the brightness value of that color. It's useful for dimming everything, for example the "Dim to 30% Brightness" preset in the Starter Pack. The "Turn On" and "Turn Off" assignment types do just what you would expect.
Most of the time, you'll probably be dealing with the "Colors" assignment type, but knowing what all the assignment types do will help you get the most out of Lightbow and your lighting setup. One other place you'll see the five assignment types is when you tap the + button in the preset list to create a new preset. For your convenience, there are buttons right in the creation panel to create the first assignment within that preset: