The easiest color combination is to pick a primary color and secondary color. Primary colors are red, blue and yellow; their secondary colors are, respectively, green, orange and purple. These pairs are opposites of one another on a color wheel, and they naturally go well together, especially when used with white. The resulting color scheme is basic but effective.
You can group warm and cool colors together to present a certain style. Warm colors are the colors between pink and yellow on a color wheel, and include red and orange. Cool colors are the other half of the color wheel, and include purple, blue and green. These combinations can make a design active and engaging or calm and relaxing.
By pairing certain colors, you can create an emotional response. For example, the colors that define Christmas are red and green, and Halloween is defined by black and orange. This principle can work for other emotions as well. A common technique is to pair three or four colors together to evoke a seasonal response, such as fall. There are no rules to evoking an emotion; the design just "feels" like what you're aiming for. Other emotion-driven color pairing examples are those that go with America, love and sunshine.
Another way to pair colors is to use tones of the same color in different brightness. Choose your base color, which is typically the most pure tone. Then select three to four tones of that color that are gradually brighter than the previous one until you reach white. This combination is simple and two dimensional but may suit your needs. It also can be easily converted to gray scale.