Path connectedness and locally path connected

One counterexample is a variant on the famous topologist's sine curve.

Consider the graph of $y = \sin(\pi/x)$ for $0<x<1$, together with a closed arc from the point $(1,0)$ to $(0,0)$:

This space is obviously path-connected, but it is not locally path-connected (or even locally connected) at the point $(0,0)$.

You should consider the opposite question, that how a space could be locally path connected, but not path connected. And this should be simple: consider the union of two open disks.

$\pi$-Base, an online version of Steen and Seebach's Counterexamples in Topology, lists the following spaces as path-connected but not locally path-connected. You can view the search result for more information about these spaces.

Alexandroff Square

Extended Topologist’s Sine Curve

The Closed Infinite Broom

The Integer Broom