In fact it probably won't be just the website we are hosting. There will be your email services too.
For now let's talk about the website. 99% of websites don't get enough traffic to require more than basic hosting. But you want the traffic you do get not to be hanging around waiting for each page to load. What sort of connection they have is relevant of course, but if your website is hosted on a very full or very busy server then your website will run slow.
Try running all the applications on your computer at the same time and see what happens... it probably gets a bit slow. Now imagine if one of those applications is trying to open a paricuarly large file. The whole computer will slow down effecting applications that have nothing to do with the one opening the large file.
Most commonly website are hosted on commercial shared hosting. So your website is sharing a computer, server, with any number of other unrelated websites. If one of those gets really busy all the other websites on that server will slow down.
How are we different? Firstly we manage our own server, based in a datacentre in the North of England. Every website on that server is managed by us. We keep the loading to a fraction of what a commercial shared server load would be. We also employ a system that manages the users if a website gets very busy so that it won't effect any other website. The system we use is NetPrecept's vAC.
Some websites have particular needs or are just very busy. These we suggest have a server to themselves, a dedicated server. This way the server can be configured to the needs and expected demands of the website.
Getting the hosting right, means the performance of the website will be right. This in turn means the best user experience and it will effect your position on Google.