Hosting Types

Shared Server
This is where ISPs lease out space on a server.  This is the least expensive method of hosting (usually $5 - $15 per month).  And for all but the busiest sites, should provide more than adequate resources.  The one negative is that ISPs have a tendancy to over subscribe resources. 

Virtual Private Server
This is the next step up.  ISPs will guarantee resources at a defined level.  You are still in a shared server environment, but now with a guarantee.  This obviously costs more (usually between $20 - $50 per month).

Dedicated Private Server
This is exactly what it sounds like.  You now have a system environment dedicated to you.  This is the most expensive hosting solution.  Pricing can be $100 to $???? per month depending on how much resource you require.  The plus side is that all the infrastructure required (internet connection, environmentals, etc) are provided as part of the cost.  The negative, you will need to have your own server administration staff. 

Web Site Hosting

Choosing a company to host your web site can be daunting task.  The first hurdle is obviously finding them.  You can look for them on the interrnet (search phrase - web site hosting). 
You can go to these sites which rate various ISPs. 

Or obviously, you can go with the one your web site developer recommends. 

For a shared server environment (which is sufficient for the vast majority of web sites), I have one that I recomend.  I have been using it for several years and have not found anything to make me want to change.  They are very reasonably priced, offer good customer support, and I have not had any problems with them over subscribing their resources.  And yes, I do monitor this segment of the industry on a regular basis.