A brain hemisphere is one of the two parts of the human brain. The brain is known to be split into the left and the right 'cerebral hemisphere'. You probably already knew this, but didn't really know what it is exactly that a brain hemisphere does.
Well, first of all the left part of your brain is basically in controlling of dealing with the things you see and think, processing information and organizing all the things you think about on a daily basis. It also helps you interpret things at a societal level. The right brain hemisphere, however, is in control of everything that you do in the present - essentially, it allows you to react to things and do things as soon as you think about it. It allows you to be you. The right half of the brain is also in control of the left half of the body, whilst the left brain hemisphere is also in control of the right half of the body.
As mentioned, the right brain hemisphere deals with everything that you experience in the present. It doesn't get involved with the processing thoughts or anything along the lines of that, but instead it allows you to have conversations, make judgments about people and things, make judgments' about things you're going to do and choose to do things.
The left brain hemisphere is really quite different. It basically processes the things that the right brain hemisphere has 'made' a person do, allowing you to recall things in your mind, consider things more carefully and make informed decisions. The left side is essentially like the computer that organizes your life and organizes the things that you do, whilst allowing you to decide what it is you want to do in the long run - not right now. In order to function as a human being, and not just a 'normal' person, we absolutely need both parts of the brain. Without it, our body would fail to work and we'd simply die.
Whilst this is true, what's particularly interesting is that some people actually rely more heavily on one side of the brain than the other. You can do this by looking at peoples' lifestyles and making informed decisions. For instance, somebody who lives a creative, intelligent, adventurous and interesting lifestyle will rely heavily on the right side of their brain. This allows them to think carefully about their lives, which often results in them being successful in their endeavors.
People who rely heavily on the left side of their brain, however, are less likely to try adventurous and daring things, and are often too 'logical' to be able to enjoy new things. Their analytical skills are second to none, and oftentimes they will find themselves assessing their life, their options, and other things that affect them to come to a decision. Relying on one side of the brain results in some very different kinds of people, but in the end, we need both halves of the brain to be able to function as human beings.