If you are looking to move your existing website to a new hosting or web programming company then you will need to have the cooperation from your existing supplier to do so. Each website has a number of files that will be stored on a server by the hosting company. Most websites also have a MySQL database which holds the information that is stored on each page or product on your site.
You may have a content management system that allows you to log in and make changes to your site and this will either be an off the shelf one or one that is bespoke and has been designed by your supplier. If the website uses an off the shelf system it is usually fairly easy to migrate it to a new supplier as long as you can get access to the files and database. If the website uses a bespoke CMS system then you may need to format the data to allow it to be uploaded in to a new CMS program.