If you want to have a presence online then you need web hosting. A hosting company is like the power company, you really don’t pay much attention to them until they are no longer there. Here in Colombia the power goes off three or four times a day and if you don’t protect your property with some kind of backup system it will destroy every electric product in your house and or office.
A bad hosting company is the same way and can destroy your online presences.
The purpose of the hosting company is to store your web pages and display them when someone types your web address into the browser. If your website is down then the person looking for your site sees nothing and if they can’t see it then neither can the search engines.
If the search engines are unable to access your website after a couple tries then they will drop the site from their index. But it gets a little more complicated than that. If the site is slow to load then it could get penalized and not rank as well as it otherwise could and site visitors could leave before the site finishes loading.
Google also uses the location of the server to decide where you want to do business. If the server is in the UK then you could very well rank well in the UK search results but not at all in the US search results.
Pick your web hosting company carefully and keep in mind that if your site is down chances are the hosting company’s site will also be down. This means you want be sending them an email about the problem, you’ll need a phone number. So if they don’t have 24 hour support you could be waiting out the weekend with no site online and no one to contact about the problem. And if they don’t have a number you’ll be waiting for them to figure out there is a problem and want be able to contact or do anything about it.