Jared Smith has provided a new tool based on many color checkers available that offers the ability to lighten and darken your color combinations slightly to find acceptable contrast levels. This new tool is at https://webaim.org/resources/contrastchecker/
The JuicyStudio Colour Contrast Analyser now redirects to the Luminosity Colour Contrast Ratio Analyser. The old Accessibility Evaluation and Repair Tools (AERT) suggested algorithm for determining colour contrast was replaced with the WCAG 2.0's luminosity contrast algorithm.
Success Criterion 1.4.3 of WCAG 2.0 requires that the visual presentation of text and images of text has a contrast ratio of at least 4.5:1. See the site for descriptions of the exceptions for large text, incidental text or images, and logotypes.