I am opening a business but another comapany already has the name registered…

No, if you are in an *at all* similar line of business, you cannot trademark the name: even though it has not been trademarked, it is already being used in commerce. Only original names, etc. can be trademarked, because they must uniquely identify the business (i.e. the source of the goods and services). However, if you and they are in very different lines of work, you may be able to trademark the name (assuming the name otherwise meets the criteria to trademarked: e.g. not generic, not descriptive, etc.), since with sufficiently different lines of business, the likelihood of customer confusion as to who is who (the thing that trademark is supposed to avoid, by identifying the source of goods or services) is minimal at best. Real life example: the "Coach" handbag company has a strong, well-known, long-standing trademark. But Triumph Learning, a  mid-sized eduational publisher, was able to trademark "Coach" as the name of a line of test preparation books despite the existence of the Coach handbag company's trademark, because high-end leather bags and school workbooks are so different there is no real chance of confusion.
On the other hand, our current President has stopped other business in the real estate field from using the word "Trump" in their business names, on the grounds that given the pre-eminence of his family and brand in that field, any other real estate-related company with "Trump" in its name would be assumed to be associated with his business. The high likelihood of confusion let him stop others from coming in that that name. 
The same principal applies even to non-trademarked names: if they are there first and do something similar to you, you can be blocked by their prior use of the name from trademarking it. If in different fields, you should be able to.