The importance of Integrality Gap I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real solution of the relaxation of the problem. Lets consider vertex cover (VC) as an example. VC can be stated as finding an optimal integer solution of the following set of linear equations: 

We have zero/one valued variables $x_v$s for each vertex $v \in V(G)$ of the graph $G$. The equations are: $0 \leq x_v \leq 1$ for $v\in V(G)$, and $1 \leq x_v+x_u$ for each edge $uv \in E(G)$. We are looking for values which will minimize $\sum_{v \in V(G)} x_v$.

The relaxation of this problem allows real values between $0$ and $1$ so the space of solutions is larger and an optimal real solution can be smaller than an optimal integer solution we want to find. Therefore we need to perform a "rounding" process on the optimal real answer obtained from linear programming to find an integer solution. The optimal integer solution will be between the optimal real solution and the result of the rounding process. IG is the ratio of an optimal integer solution to an optimal real solution and does not say anything about the rounding process. The rounding process can (in theory) completely ignore the real solution and compute the optimal integer solution directly. 


  Why are people interested in proving bounds on IG?
Integrality gaps essentially represent the inherent limits of a particular linear or convex relaxation in approximating an integer program.  Generally, if the integrality gap of a particular relaxation is $x$, then any approximation algorithm based on that relaxation cannot hope to do better than an $x$-approximation.  So at the very least, integrality gaps are of interest to algorithm designers since they suggest limitations in certain techniques.  

So why not just come up with another LP relaxation or switch to other techniques and move on?  Linear and convex programming have proven to be central to approximation algorithms; for many problems the integrality gap of a natural LP or SDP formulation is equal to the approximation ratio of the best algorithm as well as the hardness of approximation ratio.  This is just an empirical observation, but it means that proving an integrality gap may suggest much stronger consequences of an improved algorithm or lower bound.

There may be deeper and more rigorous reasons for this phenomenon.  For instance, assuming the unique games conjecture, it is known that the approximation ratio and inapproximability ratio for constraint satisfaction problems is equal to the integrality gap of a simple SDP relaxation (see Optimal Algorithms and Inapproximability Results for Every CSP? by Prasad Raghavendra)

Finally, integrality gaps represent unconditional lower bounds.  Usually, we need to rely on unproven assumptions (e.g. $P \neq NP$) if we want to make any progress in lower bounds, but for restricted models of computation, we can sometimes get away without it (see lecture notes by Luca Trevisan).  Integrality gaps, being purely geometric rather than computational, are one way of getting fairly powerful lower bounds without the baggage of extra assumptions.