When we talk about how binary search works, there are a few important points to remember. These points will help you understand why it is so efficient.

1. Time Complexity: This is where binary search really stands out. The time complexity is $O(\log n)$. This means that each time we check a middle item in a list, we can ignore half of the list. For example, if you start with a list of size $n$, checking the middle means you cut the search space in half. This makes the number of checks much smaller and faster!

2. Space Complexity: Binary search uses very little extra space. For the version that repeats steps, or iterative version, it has a space complexity of $O(1)$. This means it uses a constant amount of space, just a small amount. The version that calls itself, known as recursive, has a space complexity of $O(\log n)$. This is due to how it keeps track of its calls but is still efficient.

3. Preconditions: A vital point about binary search is that the list must be sorted before you use it. Many new users make the mistake of trying binary search on a messy, unsorted list. This can lead to wrong results. Always make sure your data is sorted first!

To sum it all up, the main points are: binary search is fast with a time complexity of $\log n$, uses little extra space, and needs the list to be sorted. Knowing these ideas will help you understand why binary search is such a popular way to search through data.