Consider a uniform space $X$ (with induced topology).
What of the following can be a subset of the other? Which of the following is always a subset of the other?
- Neighborhood of the diagonal in the topology corresponding to $X\times X$.
- The set of entourages of our uniformity.
As Daniel Fisher wrote in a comment, every entourage is a neighborhood of the diagonal. The converse fails, for example, if $X$ is the real line with the uniform structure induced by the usual metric. Specifically, $\{(x,y): y<x+e^{-x}\}$ is an open neighborhood of the diagonal, but not an entourage (because its upper boundary gets too close to the diagonal for large $x$).
If I remember correctly, though, a compact Hausdorff space always has a unique uniform structure, and the entourages of this structure are exactly the neighborhoods of the diagonal.