Milo beat me to the punch while I was writing this but I will expand on his answer.
To set your cookie an easy method would be to use jQuery Cookie.
Download the plugin and include the cookie js via wp_enqueue_script with jquery as a dependency.
For your buttons:
<input id="excerpt_view" type="button" value="Show Excerpts" />
<input id="full" type="button" value="Show Entire Posts" />
Your js to save the cookie.
jQuery(function($) {
$('#excerpt_view').click(function(e) {
$.cookie('mysite_cookie', 'true', { expires: 30 });
});
$('#full').click(function(e) {
$.cookie('mysite_cookie', null);
});
});
The PHP conditional in Milo’s answer will read the cookie. The excerpt_view button will set the cookie and cause the conditional to return true and show the excerpt. The full view button deletes the cookie which will cause the conditional to return false and show the full posts.