This color palette is closely related to analogous, but instead of using colors directly next ot each other on the color wheel, you take every other color. For example, you could choose green/yellow/orange or blue/purple/red.
Complementary colors are probably the most well known of the 7 color relationships. The palettes are comprised of two colors directly opposite the color wheel. These are most easily seen at Christmas time (red/green) and sports teams (blue/orange, purple/yellow). Using just the traditional hues from the color wheel in complementary ways can come off as cliche and unsophisticated. But, by using secondary or tertiary colors, fresh palettes can be created using complementary colors. For example, in the photo below, the "orange" of building is quite muted and desaturated, but it still really works and isn't whacking you over the head with complementary colors like the uniform of a football team.
This is similar to the complementary palette, but instead of its complement being chosen, the colors on either side of it are. For example, blue and orange are complementary colors, but instead of choosing orange, you would use blue, red-orange and orange-yellow.
By adding the third color, it can lessen some of the inherent tension and vibrancy between complementary colors and add a little nuance.
Another three color palette is created by choosing three colors that are equally distant from each other on the color wheel. Using this mix of colors, you can create vibrant color arrangments with each color playing off the others to create harmony.
In this color palette 4 colors are used. You can use what is called a rectangular tetrad or a square tetrad (shown below). With a tetrad color palette you are using two groups of complementary colors. This can quickly get out of hand because you have so many colors to work with. A simple way to reign things back into control is to choose one dominant color and let the rest act as accent colors. While maybe a bit forced, you can see the tetrad used in the Van Gogh painting below. The painting was intentionally created to be harsh and unsettling. He is using color to his advantage by using complementary colors to create a vibrating tension between the colors.
I wanted to include neutrals here as well since they are such a common palette to see. While anything other than black, white and gray are still technically colors, most of the saturation has been removed in neutrals. Neutrals tend to look sophisticated and are calming to our senses. By using them carefully and being aware of how warm and cool neutrals act together, they are a pretty easy color palette to work with. This is why most houses have beige walls (it goes with everything so I hear).
Neutrals can work within a lot of different spaces. The differences between a neutral palette working beautifully like below and a neutral palette just being boring are subtle and sometimes difficult to achieve. One easy way to liven up a bland neutral palette is by introducing an accent color. In the room below, the yellow glow of the lamp adds interest and warmth to an otherwise gray and brown room.
The mystery of why some color combinations hum and why others don't quite work may be difficult to sort out at times, but with a little analysis, the mystery can be removed. The only way to get better at choosing color palettes, whether it's for a few acoustic panels to match your existing office furniture, or whether you're designing a whole building or painting a painting is to practice. Sometimes you can accidentally hit on a color palette that works, but being consciously aware of what you are doing and where those colors fall on the color wheel can increase your odds of success. As you go throughout your day, try to be consciously aware of the colors you see around you and place them mentally on the color wheel.
Another place to go for some inspiration is of course, the internet. Below are three web-based, free and easy color palette tools that are useful for creating, exploring and tweaking color palettes. You can then export or record the colors used and apply them to whatever you are creating!
Coolors - Self described as "the super fast color schemes generator for cool designers!" It's a really easy to use and beautifully designed tool.
Colour Lovers - This site has a lot of inspirational color palettes, and you browse by categories. There is also information on trends in various markets ranging from branding to fashion.
Adobe Color - From Adobe we have what I would consider the most technical of the sites listed here. You have a lot of controls to tweak colors very precisely. One nice feature is other colors in the palette will shift automatically to corresponding values depending on whether you chose Analagous, complementary, etc... as your color rule.