If you search on Google for the term "joomla can't login after moving to new server" - or something similar, you will get more, than 1 million hits for most of the combinations.
So, you aren't alone. And there are plenty of suggestions, on what might gone wrong and how you might find that, and there are verious fixes. But what if you moved the site to another domain? From development server to it's final place? This is a subject wich is rarely touched by those advices.
But this is what can be your problem. If otherwhise the site looks/works OK, this is what you need to cehck first.
There are two settings/variables in the configuration.php wich can have this side effect. One of them is as old as Joomla is, it's the LiveSite variable:
public $live_site= '';
Check to be empty - as in the example above - or to match your site's preferred domain - with or withouth the www. prefix, as you have it set in .htaccess, in Goole Webmaster Tools or any SEF plugins/components you are using (sh404SEf for example, but not only).
The other potential culprit is a newer one, is the Cookie Domain variable. Is safe to set it for security purposes, it's a way to prevent session hijacking type of attacks for example, but can wreak havok in your cookie handling if the setting is wrong. Check it, and configure it as you need it!
public $cookie_domain = 'mynewdomain.com';
Small and simple things as they are, these settings are often forgotten. Take care of them - and they will help you to have a better performing Joomla site!