What are approximation algorithms and approximation ratios?

Approximation algorithms find near-optimal solutions to NP-hard problems in polynomial time with guaranteed quality. Approximation ratio r means solution is within factor r of optimal. Examples: Vertex Cover has 2-approximation (take both endpoints of each edge). Set Cover has O(log n)-approximation (greedy). TSP metric has 1.5-approximation (Christofides). PTAS (Polynomial Time Approximation Scheme) achieves (1+epsilon) ratio for any epsilon. Inapproximability results show limits.