For the sake of answering this question, I'm going to draw from examples of the United States government. As you used "our" in your question, it's a bit unclear if that is what you need, but the basic principles should be largely the same no matter what government we are dealing with.
A faction within a government is very much what it sounds like. It is a group of people within a larger group that are united in regards to one (or several) issues. Political factions are basically smaller groups inside of political parties that are far more focused and specific.
For instance, the Democratic Party has a wide, relatively vague set of beliefs that trend more liberal than the Republican Party. However, some Democrats might disagree on a controversial issue like abortion. Some may believe that abortion is a common right, while others may think that you should not be allowed to have an abortion after a certain number of weeks. A group of Senators could group together and actively work towards an agreed-upon goal making abortion illegal after that time period that they think is correct for the situation. This would make them a faction, as they are a smaller group working to achieve a specific ideological goal.
Factions are pretty important in politics, and they can have both negative and positive effects. On the negative side, they can end up splitting a party across issues and making it harder to get anything done. On the other hand, it's easier to get laws passed and change to actually happen if you have multiple people working together, rather than just working as a single individual.
Good question, and I hope this helps! Good luck.