It's quite simple actually:
var myArray = new Array("a", "b", "c", "b", "a");
var newObject = {};
// Iterate over the array
for(var i = 0; i < myArray.length; i++){
// If the new object already contains the key (e.g. a, b, or c), increment value by one
if(myArray[i] in newObject){
newObject[myArray[i]]++;
} else {
// Otherwise add a key (e.g. a, b, or c) to the object and assign 1 to it (first occurence)
newObject[myArray[i]] = 1;
}
}
// Write the resulting object to console
window.console && console.log(newObject);
newObject
contains a list of keys (a,b,c) and values (number of occurrences of each key). You can than use that data to output it in any format you like, but that's left up to you as an excercise.