Should I make HTML Anchors with 'name' or 'id'?

You shouldn’t use <h1><a name="foo"/>Foo Title</h1> in any flavor of HTML served as text/html, because the XML empty element syntax isn’t supported in text/html. However, <h1><a name="foo">Foo Title</a></h1> is OK in HTML4. It is not valid in HTML5 as currently drafted.

<h1 id="foo">Foo Title</h1> is OK in both HTML4 and HTML5. This won’t work in Netscape 4, but you’ll probably use a dozen other features that don’t work in Netscape 4.

According to the HTML 5 specification, 5.9.8 Navigating to a fragment identifier:

For HTML documents (and the text/html MIME type), the following processing model must be followed to determine what the indicated part of the document is.

1. Parse the URL, and let fragid be the <fragment> component of the URL.
2. If fragid is the empty string, then the indicated part of the document is the top of the document.
3. If there is an element in the DOM that has an ID exactly equal to fragid, then the first such element in tree order is the indicated part of the document; stop the algorithm here.
4. If there is an a element in the DOM that has a name attribute whose value is exactly equal to fragid, then the first such element in tree order is the indicated part of the document; stop the algorithm here.
5. Otherwise, there is no indicated part of the document.

So, it will look for id="foo", and then will follow to name="foo"

Edit: As pointed out by @hsivonen, in HTML5 the a element has no name attribute. However, the above rules still apply to other named elements.