A well-connected topological space is one that satisfies sufficiently strong local connectivity assumptions. These are usually defined in terms of a basis of open sets or a basis of neighbourhoods of the space. Note that this term is not universally defined, but is a placeholder for possibly more complicated adjectives that depend on the situation at hand.
A locally path-connected space has the nice property that path-components are the same as components. It is also true in this case that $X^I \to X\times X$ is an open map.
A locally path-connected and semi-locally simply-connected space admits a universal covering space.
The path fibration $P X \to X$ of a path-connected space admits local sections if and only if $X$ is locally relatively contractible.
CW complexes are the 'most' well-connected spaces, having a basis of open sets which are themselves contractible as spaces.