Tufte suggests six fundamental principles of design: show comparisons, show causality, use multivariate data, completely integrate modes (like text, images, numbers), establish credibility, and focus on content. Each should be geared towards fully embracing the goals you defined for a given data display. For each principle, we outline examples of how to apply it to improve your visualizations.

Show comparisons
One of the major examples Tufte uses in showing comparisons looks at Charles Joseph Minard's map of Napoleon's march to and return from Russia. Minard shows the size of the French army by the width of the line. This makes plain how the army rapidly reduced in size as it marched east, and also makes the comparison between the size of the army that departed France and the size of the army that returned clear. 

Edward Tufte, Beautiful Evidence (2006), 122

Show causality
In the example above, the goal was to show comparisons. Another possible goal is to show causality. In the above example, we can see causaulity in the black returning line of soldiers and the graph of temperature at the bottom of the chart. Minard shows how without even engaging in battle, the march itself killed thousands due to freezing temperatures.

Another example shows the influence of music groups on one another over a twenty year period, while a second shows the transmission of SARS.
Reebee Garafalo from Edward Tufte, Visual Explanations (1997) p 90 and Beautiful Evidence (2006) p 78

Use multivariate data
The idea of using multivariate data may seem straightforward, but fully embracing this idea can lead to spectacular results. The example below, which outlines the lifecycle of the Japanese beetle, contains information on time, space, physical appearance, behavior, and interaction with its ecosystem. As this example shows, every relevant type of information should be included.

Japanese beetle lifecycle
L. Hugh Newman, Man and Insects (London, 1965), 104-105

Complete integrate modes
To improve displays of data, it is important to completely integrate text, images and numbers. It is also important to include any information or data that is relevant. This is particularly important in the inclusion of explanatory text on figures. In the Minard example, this means an integration of the chart of the map and army size, but also text information about location, numerical information about the temperature, etc.

In the examples showing various information about animal sizes, we can also see how an effective integration of text can significantly improve readability of the chart, but that integration of pictures can make it even more meaningful.
Edward Tufte, Beautiful Evidence (2006) p 121

Establish credibility
Tufte argues repeatedly that including source information is one of the most important aspects of creating a convincing visual display or a convincing presentation. Allowing the viewers to access the source material will give them confidence in your results. The Minard chart is a classic example of this, prominently featuring both the author and his references in the text introduction at the top of the chart.

Focus on content
Above all else, Tufte argues that you must focus on the content to make an effective chart. That means, as outlined in the section on best practices, minimizing the chart architecture wherever possible. It also means that if you find a chart that is more "chartjunk" than content, you should research more and find more valuable content before you create or present your data. Even in the simplest cases, it is possible to display a rich world of information. In the example below, a simple bird book entry on Jays, the writer presents an abundance of information. There is information about common and scientific names, sizes and relative sizes, coloration, appearance in flight, environment and migration patterns, and even the sound of their calls. This kind of display driven by content will make the most "beautiful evidence."