Well, maybe your organizational structure does not have to change at all. DevOps has to start with a cultural change: openness, empathy, and collaboration are values that need to be encouraged. But still, changing your organizational structure might help to accelerate this.
Traditionally, developers and operators are often organized in disparate teams or even different departments—organized in teams with people that have a similar skill set and responsibility. A common change to organizations is changing this structure, by pivoting and organizing teams behind a common goal, a single product, or a group of features, for example.
Now you will need teams with different skill sets and responsibilities, teams most likely with developers and operators. It is important to realize that forcing such a change upon these people might not be the best way forward. Often, it works best to start with changing the culture, encouraging cooperation, and then this organizational change might come about in a natural way.
Finally, it is important to recognize one anti-pattern at this point. Some companies are trying to implement DevOps by hiring specialized DevOps engineers and positioning them between development and operations, interacting with both. While this, at first, might seem like a good idea, this goes against the DevOps values. If you do this, you are not breaking silos down, but you are adding a third one. You are not decreasing the number of hand-offs, you are most likely increasing them. Also, collaboration between developers and operations is often not enhanced by separating them using another organizational structure and you might not see any increase in value to your end users at all.
Now that you know what DevOps is and you have a clear understanding of how you can form a DevOps team, it is time to explore how to start achieving your goals.