Migration of Craft CMS from one host to another
You need a copy of the database which you can either export via Craft's Control Panel or via phpMyAdmin or a similar database management tool.
You need these folders: 


craft/templates
craft/translations (if it's multi-language)
craft/plugins
craft/config
public


They might be called differently, I am writing this as if the previous developer followed the installation instructions. 

Upload a fresh copy of Craft to the new server. 

Import the database. 

Replace the stock folders with the ones you got from the other install. 

Make sure db-credentials are correct in craft/config/db.php. Make sure all paths are correct in craft/config/general.php. 

Make sure the folders that need to be writeable are writeable.

Generally speaking your site should be running now.