Most standard plans (starting at ~$5 a month) come with php and mysql support by default.
Usually the hosting company puts a link to some management tool in your admin area, most of the time it's phpMyAdmin. You can use that to manage your database, should the need arise.
Unless you're planning on creating your shop from scratch (not recommended unless you are a pro), go for Wordpress and its WooCommerce (shop) plugin. It's all free, and the shop is absolutely professional.
Your database only stores text information. You will need a lot more than just a MySQL database. To create your own custom store, you will need to know PHP, HTML, CSS, and MySQL.
Most people don't have time to learn this so they lean towards using a web application such as WordPress. Its an open source program that has an admin interface to create a website. You will need a hosting account that supports php and MySQL. I know a website called webneto.com that offers this that is pretty reliable.
I understand your concerned about having a database. However, if you don't have a website, then your database is useless. Your database won't do anything if you don't have a website.
Depending which web hosting package you purchase, the web hosting company might install what you need, for your website to work, or you will have to do it yourself.
I have seen shopping carts before. I think using WordPress is the best option you have to make an online shopping website. You might be able to make one from scratch, but it could take you most of your life to figure out how to make an online shopping website from scratch.
I know WordPress uses MySQL, so you would get what you want. I think your best option is, for you to surf the Internet to find guides that would benifit you to learn how to setup, and use WordPress.
Learn php and MySQL. Or you can use a website builder like ultimatewb.
Download the free script "mini e-commerce" from http://webprosonly.com
I want to create an online shopping website, where customers can log in and make orders. I'd definitely need a database. I have chosen to use MySQL. What i'm trying to understand is where is my database going to be stored? How will i publish the database to the web?
