Browsers do this automatically with the help of the several caching mechanisms that HTTP provides. The two mechanisms most obviously useful for determining whether a page has changed is the concept of Entity Tags and the Last Modified HTTP header. These mechanisms allow the browser to make conditional requests to a web site, eg. fetch a page only if it has been changed.
Quoting RFC 2616 on HTTP 1.1:
3.11 Entity Tags
Entity tags are used for comparing two or more entities from the same
requested resource. HTTP/1.1 uses entity tags in the ETag (section
14.19), If-Match (section 14.24), If-None-Match (section 14.26), and
If-Range (section 14.27) header fields. The definition of how they
are used and compared as cache validators is in section 13.3.3. An
entity tag consists of an opaque quoted string, possibly prefixed by
a weakness indicator.
The key point here is that the ETag is a cache validator. If a browser has a cached version of a page (called a resource in the RFC), it can use the ETag to determine whether the cached page is still valid, ie. if the page hasn't changed on the server.
And about the modification date:
14.25 If-Modified-Since
The If-Modified-Since request-header field is used with a method to
make it conditional: if the requested variant has not been modified
since the time specified in this field, an entity will not be
returned from the server; instead, a 304 (not modified) response will
be returned without any message-body.
The key point here is that the server may know when a page has been modified, and may then inform the client.
If you open a HTTP monitor (such as Fiddler for Windows) and watch your browser communicate with web sites, you'll see the use of these mechanisms first-hand when the browser makes conditional requests.
To specifically address your question about the Last Modified header, this header in itself won't work for the majority of pages you'll find. But in combination with the ETag it can get you started.