A priority map is very similar to a sorted map, but whereas a sorted map produces a
sequence of the entries sorted by key, a priority map produces the entries sorted by value.
In addition to supporting all the functions a sorted map supports, a priority map
can also be thought of as a queue of [item priority] pairs.  To support usage as
a versatile priority queue, priority maps also support conj/peek/pop operations.

The standard way to construct a priority map is with priority-map:
user=> (def p (priority-map :a 2 :b 1 :c 3 :d 5 :e 4 :f 3))
#'user/p
user=> p
{:b 1, :a 2, :c 3, :f 3, :e 4, :d 5}

So :b has priority 1, :a has priority 2, and so on.
Notice how the priority map prints in an order sorted by its priorities (i.e., the map's values)

We can use assoc to assign a priority to a new item:
user=> (assoc p :g 1)
{:b 1, :g 1, :a 2, :c 3, :f 3, :e 4, :d 5}

or to assign a new priority to an extant item:
user=> (assoc p :c 4)
{:b 1, :a 2, :f 3, :c 4, :e 4, :d 5}

We can remove an item from the priority map:
user=> (dissoc p :e)
{:b 1, :a 2, :c 3, :f 3, :d 5}

An alternative way to add to the priority map is to conj a [item priority] pair:
user=> (conj p [:g 0])
{:g 0, :b 1, :a 2, :c 3, :f 3, :e 4, :d 5}

or use into:
user=> (into p [[:g 0] [:h 1] [:i 2]])
{:g 0, :b 1, :h 1, :a 2, :i 2, :c 3, :f 3, :e 4, :d 5}

Priority maps are countable:
user=> (count p)
6

Like other maps, equivalence is based not on type, but on contents.
In other words, just as a sorted-map can be equal to a hash-map,
so can a priority-map.
user=> (= p {:b 1, :a 2, :c 3, :f 3, :e 4, :d 5})
true

You can test them for emptiness:
user=> (empty? (priority-map))
true
user=> (empty? p)
false

You can test whether an item is in the priority map:
user=> (contains? p :a)
true
user=> (contains? p :g)
false

It is easy to look up the priority of a given item, using any of the standard map mechanisms:
user=> (get p :a)
2
user=> (get p :g 10)
10
user=> (p :a)
2
user=> (:a p)
2

Priority maps derive much of their utility by providing priority-based seq.
Note that no guarantees are made about the order in which items of the same priority appear.
user=> (seq p)
([:b 1] [:a 2] [:c 3] [:f 3] [:e 4] [:d 5])
Because no guarantees are made about the order of same-priority items, note that
rseq might not be an exact reverse of the seq.  It is only guaranteed to be in
descending order.
user=> (rseq p)
([:d 5] [:e 4] [:c 3] [:f 3] [:a 2] [:b 1])

This means first/rest/next/for/map/etc. all operate in priority order.
user=> (first p)
[:b 1]
user=> (rest p)
([:a 2] [:c 3] [:f 3] [:e 4] [:d 5])

Priority maps support metadata:
user=> (meta (with-meta p {:extra :info}))
{:extra :info}

But perhaps most importantly, priority maps can also function as priority queues.
peek, like first, gives you the first [item priority] pair in the collection.
pop removes the first [item priority] from the collection.
(Note that unlike rest, which returns a seq, pop returns a priority map).

user=> (peek p)
[:b 1]
user=> (pop p)
{:a 2, :c 3, :f 3, :e 4, :d 5}

It is also possible to use a custom comparator:
user=> (priority-map-by > :a 1 :b 2 :c 3)
{:c 3, :b 2, :a 1}

Sometimes, it is desirable to have a map where the values contain more information
than just the priority.  For example, let's say you want a map like:
{:a [2 :apple], :b [1 :banana], :c [3 :carrot]}
and you want to sort the map by the numeric priority found in the pair.

A common mistake is to try to solve this with a custom comparator:
(priority-map 
  (fn [[priority1 _] [priority2 _]] (< priority1 priority2))
  :a [2 :apple], :b [1 :banana], :c [3 :carrot])

This will not work!  In Clojure, like Java, all comparators must be total orders,
meaning that you can't have a tie unless the objects you are comparing are
in fact equal.  The above comparator breaks that rule because
[2 :apple] and [2 :apricot] tie, but are not equal.

The correct way to construct such a priority map is by specifying a keyfn, which is used
to extract the true priority from the priority map's vals.  (Note: It might seem a little odd
that the priority-extraction function is called a *key*fn, even though it is applied to the
map's values.  This terminology is based on the docstring of clojure.core/sort-by, which
uses `keyfn` for the function which extracts the sort order.) 

In the above example,

user=> (priority-map-keyfn first :a [2 :apple], :b [1 :banana], :c [3 :carrot])
{:b [1 :banana], :a [2 :apple], :c [3 :carrot]}

You can also combine a keyfn with a comparator that operates on the extracted priorities:

user=> (priority-map-keyfn-by 
          first >
          :a [2 :apple], :b [1 :banana], :c [3 :carrot])
{:c [3 :carrot], :a [2 :apple], :b [1 :banana]}

 

All of these operations are efficient.  Generally speaking, most operations
are O(log n) where n is the number of distinct priorities.  Some operations
(for example, straightforward lookup of an item's priority, or testing
whether a given item is in the priority map) are as efficient
as Clojure's built-in map.

The key to this efficiency is that internally, not only does the priority map store
an ordinary hash map of items to priority, but it also stores a sorted map that
maps priorities to sets of items with that priority.

A typical textbook priority queue data structure supports at the ability to add
a [item priority] pair to the queue, and to pop/peek the next [item priority] pair.
But many real-world applications of priority queues require more features, such
as the ability to test whether something is already in the queue, or to reassign
a priority.  For example, a standard formulation of Dijkstra's algorithm requires the
ability to reduce the priority number associated with a given item.  Once you
throw persistence into the mix with the desire to adjust priorities, the traditional
structures just don't work that well.

This particular blend of Clojure's built-in hash sets, hash maps, and sorted maps
proved to be a great way to implement an especially flexible persistent priority queue.

Connoisseurs of algorithms will note that this structure's peek operation is not O(1) as
it would be if based upon a heap data structure, but I feel this is a small concession for
the blend of persistence, priority reassignment, and priority-sorted seq, which can be
quite expensive to achieve with a heap (I did actually try this for comparison).  Furthermore,
this peek's logarithmic behavior is quite good (on my computer I can do a million
peeks at a priority map with a million items in 750ms).  Also, consider that peek and pop
usually follow one another, and even with a heap, pop is logarithmic.  So the net combination
of peek and pop is not much different between this versatile formulation of a priority map and
a more limited heap-based one.  In a nutshell, peek, although not O(1), is unlikely to be the
bottleneck in your program.

All in all, I hope you will find priority maps to be an easy-to-use and useful addition
to Clojure's assortment of built-in maps (hash-map and sorted-map).