Steemit's Reputation System works an indicator of a user's standing on the blockchain.
Every new user starts with a 25 reputation score. This score can increase or decrease based on the number and weight of the upvotes and downvotes you receive. So, if you write a post that others find value in, they are likely to upvote you and help increase your reputation. The opposite is also true, and people can downvote you and decrease your reputation.
@sndbox actually wrote a simple illustrated guide about the Steemit reputation system, which you can read here: https://steempeak.com/toolkit/@sndbox/what-is-a-steemit-reputation-score-an-illustrated-guide.
Now, without getting hung up on the reputation score, there are a few things that every Steemian should keep in mind as far as the reputation system is concerend.
First, it's important to note that the reputation system has no impact on your vote value. No amount of increase or increase in your reputation score will change that.
And second, while it is not important to have a high reputation, it is important to avoid getting a low reputation. As far as your activity is concerned, there is no real advantage to a high reputation. Granted, some would be more likely to follow and upvote you if you have a high reputation, but that is becoming less true, everyday. However, a low reputation is another matter entirely. Because once your reputation becomes dangerously low, your posts and comments will be automatically hidden from view, affecting your visibility and preventing others to interact with you. So, this is something that everyone should keep in mind when thinking about the reputation system.
Hope this helps.