Understanding algorithm complexity is really important when we want to figure out how well algorithms work. There are two main types of complexity we need to know about: time complexity and space complexity.

Time Complexity
Time complexity is all about how long an algorithm takes to run depending on how big the input is. We usually call the size of the input $n$. This helps us see how the time needed changes when the input gets bigger. Here are some common types of time complexity:

• Constant Time: $O(1)$ - The time stays the same no matter how big the input gets.
• Logarithmic Time: $O(\log n)$ - The time increases slowly as the input size grows, like with a binary search.
• Linear Time: $O(n)$ - The time goes up steadily as the input size gets bigger, like with a simple loop.
• Quadratic Time: $O(n^2)$ - The time goes up quickly as it gets bigger, because it relates to the square of the input size, like in nested loops.
• Exponential Time: $O(2^n)$ - The time doubles every time we add a new element, seen in some recursive problems.

Space Complexity
On the flip side, space complexity tells us how much memory an algorithm needs based on the input size. It looks at both the extra space it uses and the space that the input itself takes up. Here are the main types of space complexity:

• Constant Space: $O(1)$ - The algorithm uses the same amount of space no matter how big the input gets.
• Linear Space: $O(n)$ - The memory needed goes up steadily as the input size increases.
• Logarithmic Space: $O(\log n)$ - The amount of memory used increases slowly as the input size grows.

Both time and space complexities are super helpful when we're designing and choosing algorithms. They help us make sure that algorithms run quickly and use memory wisely. Knowing these ideas is really important for making algorithms better in data structures.