What a valid sitemap needs
A sitemap is an XML file listing the pages you want found. It must be well-formed XML, start with <urlset> (or <sitemapindex> for a list of sitemaps) in the namespace http://www.sitemaps.org/schemas/sitemap/0.9, and give every page as a full, absolute address in a <loc>. One file holds at most 50,000 URLs and 50 MB uncompressed; bigger sites split into several files and list them in an index.
What the checker looks at
The HTTP status and content type, whether the XML parses, the root element and namespace, missing or relative URLs, URLs on a different host (ignored unless that site is verified too), duplicates, and whether <lastmod> dates follow the W3C format. It also checks that robots.txt points at the sitemap, which is how crawlers you never submitted it to find it.
lastmod, priority and changefreq
Google uses <lastmod> when it is consistently accurate - so set it to the date the content really changed, not the date the sitemap was built. It ignores <priority> and <changefreq> entirely.
Then check that your robots.txt allows those pages with the robots.txt validator, and that old URLs in it do not redirect with the redirect checker - a sitemap should list final addresses only.
