I have recently wanted to start up a design company/photography business. I have chosen my name but found out someone else online was using it. Their website says copyright 2016. Another website I looked (different name) at said copyright 2016 all rights reserved. Is there a difference? Can I still use my original name or will I have to change my business name entirely?
Business names are commonly regulated and you will need to check the regulations in your jurisdiction. For example, companies in Australia are registered nationally by ASIC, however, business names are registered state by state (however, ASIC maintains a consolidated database). In Australia it is illegal for a person (including a company) to conduct business under a name that is not their own. However, if your name is "John Smith" then you can conduct business under that name irrespective of how many other John Smith's are doing the same.
Here is a business name search for "ABC" - as you can see, subtle differences ("ABC" vs "A.B.C") allow similar but distinct business (or company) names to be registered.
Notwithstanding if your name is distinct enough to allow registration, using it may still infringe another person's trade mark. A trade mark is infringed when you use someone else's trade mark in a way that could cause confusion between your goods and services and theirs. You may have a perfectly legitimately registered business name but if you do not sufficiently distinguish your goods and services from theirs you cannot use it. In your case, you are talking about establishing a "a design company/photography business"; if the person you are concerned about sells fruit, you are probably not going to infringe their trade mark. Similarly, if your operations are sufficiently removed geographically that there is minimal chance of confusion you should be fine.
Copyright is a completely different concept and has to do with artistic or literary expression. They are asserting that they have copyright in the content (literary) and appearance (artistic) of their web site. Unless you copy that you have no issues with copyright. No one has copyright in a name because names lack the necessary threshold of literary creativity.