A good application of priority queues is finding the shortest path in a graph. A common algorithm for this is Dijkstra’s algorithm.
Edsger Dijkstra was a Dutch computer scientist who researched many fields. He is credited for his work in physics, programming, software engineering, and as a systems scientist. His motivation for this algorithm in particular was to be able to find the shortest path between two cities.
What is the shortest way to travel from Rotterdam to Groningen, in general: from given city to given city? It is the algorithm for the shortest path, which I designed in about twenty minutes. One morning I was shopping in Amsterdam with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path.  Edsger Dijkstra, Communications of the ACM 53 (8), 2001.
His original algorithm was defined for a path between two specific cities. Since its publication, modifications have been made to the algorithm to find the shortest path to every node given a source node.
DIJKSTRAS(GRAPH, SRC)
SIZE = size of GRAPH
DISTS = array with length equal to SIZE
PREVIOUS = array with length equal to SIZE
set all of the entries in PREVIOUS to none
set all of the entries in DISTS to infinity
DISTS[SRC] = 0
PQ = minpriority queue
loop IDX starting at 0 up to SIZE
insert (DISTS[IDX],IDX) into PQ
while PQ is not empty
MIN = REMOVEMIN from PQ
for NODE in neighbors of MIN
WEIGHT = graph weight between MIN and NODE
CALC = DISTS[MIN] + WEIGHT
if CALC < DISTS[NODE]
DISTS[NODE] = CALC
PREVIOUS[NODE] = MIN
PQIDX = index of NODE in PQ
PQ decreasekey (PQIDX, CALC)
return DISTS and PREVIOUS
Aside from just finding routes for us to travel, Dijkstra’s algorithm can accommodate for any application that can have an abstraction to finding the shortest path. For example, the following animation shows how a robot could utilize Dijkstra’s algorithm to find the shortest path with an obstacle in the way. In this example, each node could represent one square foot of floor space and the edges would represent those spaces that are adjacent. In this scenario, we would most likely not have an associated edge weight. If the robot were traversing on a rugged terrain, then we could have the weights represent the difficultly of passing through the terrain from one space to the other.
Another practical abstraction is in network routing. In this simplified abstraction, nodes would be routers or switches and the edges would be the physical links between them. The edge weights in this case would be the cost of sending a packet from one router to the next. Dijkstra’s algorithm is actively used in protocols such as Intermediate System to Intermediate System (ISIS) and Open Shortest Path First (OSPF).

Shiyu Ji, CC BYSA 4.0 https://creativecommons.org/licenses/bysa/4.0, via Wikimedia Commons, https://upload.wikimedia.org/wikipedia/commons/e/e4/DijkstraDemo.gif ↩︎

Subh83, CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Dijkstras_progress_animation.gif ↩︎