The Law
Section 508 Part 1194.22 Subpart d states:
Documents shall be organized so they are readable without requiring an associated style sheet.
What to Do
Pages should make use of HTML tags to designate information where applicable. Use H1 for primary headings, P for paragraph text, STRONG for bold text, etc... Making use of HTML tags ensures that users with alternate style sheets or screen readers can still navigate a page and read it in a logical manner. Do not, for example, use <span style="font-weight: bold;">bold</span>.
A good way to check this is to disable all user stylesheets (with the Web Developer Toolbar for Chrome or Firefox, for example) and see if the page is still displayed in a logical manner.
Page Layout
Page layout should be accomplished using a separate style sheet. Using tables for page layout is a deprecated practice. It is common to use <div> elements to create blocks of text that can be positioned and styled with a style sheet.