Every color in the spectrum has a complementary color -- one that's the direct opposite by virtue of sharing no common colors. In red's case, this color is green, though the holiday associations of this pairing tend to limit its application. Green is technically the color that most contrasts with red, making it useful as an accent in primarily red designs. Of course, swathes of green and red are completely appropriate for holiday designs such as greeting cards.
All the colors in the rainbow, when arranged in a circle, form what's called a color wheel in the world of art and color theory. This wheel is extremely useful in creating color schemes; instructions for making one can be found at Artsparx.com. After you've created the wheel, drawing an equilateral or isosceles triangle between any three points will connect three colors that produce a harmonious color scheme. When red is connected via triangle, for example, it touches yellow and blue.
When yellow and blue are teamed up with red, all three together create an invigorating, contrasting color scheme that still works harmoniously. Yellow and blue both share attributes with red, meaning that they are not as jarring as complementary colors, yet still different enough to catch the eye and impart red's invigorating trademark. In this specific setup, blue and yellow are the other two primary colors, apart from red. Together, any color can be created from these three, and because of this relationship, the colors work well together. This is a special case; most triadic color combinations are made up of colors that can be created from a common root color. This means that the colors actually share elements of a color, and as they are all derived from the same source, they all work harmoniously. Red, yellow and blue make up a trio of pure, bright colors. Although they do not share a root color, their relationship as primary colors acts as their root connection. This makes this color scheme perfect for powerful designs intended to excite the mind, arouse the eye and spark the imagination.
White and black can be used with red in any design. They both provide the same degree of visual contrast, though black tends to make the red it works with seem darker and white makes red seem lighter. Red also stands out remarkably well when paired with white, drawing the eye directly to the red parts of an object. Paired with black, red can seem dark and ominous. Since neither color clashes with red or each other, you can use both.