NP-Complete problems are a class of computational problems that are both difficult to solve and verify. If a solution to an NP-Complete problem is provided, it can be checked quickly to confirm its correctness. However, finding that solution in the first place can take an impractical amount of time as the size of the problem grows. These problems are significant in computer science because they help researchers understand the limits of what can be efficiently computed. If any NP-Complete problem can be solved quickly, it would imply that all problems in the NP class can also be solved quickly, leading to major implications in fields like cryptography and optimization.