Using Powershell, how can i count the occurrence of each element in an array?

Joey's helpful answer provides the crucial pointer: use the Group-Object cmdlet to group input objects by equality.
(To group them by one ore more of their property values instead, use -Property).

Group-Object outputs [Microsoft.PowerShell.Commands.GroupInfo] objects that each represent a group of equal input objects, whose notable properties are:

• .Values ... the value(s) that define the group, as a [System.Collections.ArrayList] instance (which has just 1 element in the case at hand, since the input objects as a whole are used to form the groups).

• .Count ... the count of objects in that group.

If, as in this case, there is no need to collect the individual group members as part of each group,
-NoElement can be used for efficiency.

You're free to further process the group objects as needed; to get the specific output format stipulated in your question, you can use Select-Object with a calculated property.

To put it all together:

PS> 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 5, 5 | Group-Object -NoElement |
      Select-Object @{ n='Element:Count'; e={ '{0}:{1}' -f $_.Values[0], $_.Count } }

Element:Count
-------------
1:3
2:2
3:1
4:4
5:2

You can adjust the output and format it as you like:

PS> $ht= 1,1,1,2,2,3,4,4,4,4,5,5 | Group-Object -AsHashTable -AsString
PS> $ht

Name                           Value
----                           -----
2                              {2, 2}
4                              {4, 4, 4, 4}
5                              {5, 5}
1                              {1, 1, 1}
3                              {3}


PS> $ht['1']
1
1
1

You can use the Group-Object cmdlet:

PS> 1,1,1,2,2,3,4,4,4,4,5,5 | group

Count Name                      Group
----- ----                      -----
    3 1                         {1, 1, 1}
    2 2                         {2, 2}
    1 3                         {3}
    4 4                         {4, 4, 4, 4}
    2 5                         {5, 5}

If you want a hashtable for the items and their counts you just need a little ForEach-Object after it:

$array | group | % { $h = @{} } { $h[$_.Name] = $_.Count } { $h }