You need to store the files that make your website work (images, buttons, codes, etc.) in an internet server with a fixed address and always-on fast internet connection so that they can be displayed anytime for everyone that visits your website, This is what's called hosting.
When a web designer tells you that you can use your own host they are assuming you already have a server or someone that hosts your files online and basically what they are selling you is only the domain name and not a hosting service, so you need to find a Hosting Provider (like us) to store your files.
Hosting in a nutshell is renting space for your website to live on the internet. This saves the expense and hassle of getting a fixed IP address of your own and having your own webserver which you would need to manage, maintain and upgrade from time to time in your own office with a fast internet connection.
For an exceptional base for your website, we strongly reccommend JOOMLA CMS (content management system)