In fact it is a bit more complicated than just an American or British accent. There are lots of different accents in America and Britain. In Britain the accent in York is different from the accent in the south of England. Likewise, the New York accent is completely different to that in the Deep South of the USA. And there are many, many more examples. If you have a native speaker teaching you, then ask your teacher which part of the USA or Britain they are from and which accent they speak with.It will probably make for an interesting conversation. You will probably find that other people will tell you "Oh you must have a teacher from... I can hear the accent when you speak." Plus then you also have to think about other versions of English, such as Canadian, Australian, South African, Indian, etc. And then each of these have different choices of words and dialects. AND then there is the fact that English is spoken by a huge number of non-native speakers with many different accents.
What does this mean if you are learning English? Well, firstly don't worry if you don't understand some accents or dialects - often we native speakers find it difficult to understand each other BUT if both people are willing to try to communicate they will adapt the way they speak to make this possible. If the other person does not want to adapt that is not a failure on your part. So, don't be too hard on yourself if you find a conversation difficult. Secondly, try to listen to as many different accents as you can in order to improve your listening skills, don't just choose one to copy. And thirdly, in fact there is no need to try and copy a British or American accent...we are used to people speaking English with accents that are partly Indonesian, Russian, Chinese, German, Dutch, Spanish, etc, etc. You just need to have clear-enough pronunciation to communicate with all these other people!