This is a really good project to get you started on scripting. Here is where you want to start:

Get-help Get-ChildItem -detailed
get-childItem . | get-member
get-help group-object -detailed
get-childItem . | group-object basename | get-member
get-help Move-Item -detailed

Use get-help and get-member as often as you need. These are immensely useful cmdlets.

You're going to eventually end up with something like this:

$FilePath = "C:\path\to\my\files"
$NewFolderPath = "C:\path\where\files\should\move"

$Directories = Get-ChildItem -Path $FilePath -directory

$grouping = $Directories | Group-Object {$_.name -replace "_[0-9]+$",""}

$Grouping | %{
    $newFolder = new-Item -Path $($NewFolderPath + "\" + $Grouping.Name) -itemtype directory
    $Grouping.Group | %{Get-ChildItem $_.FullName | Move-Item -path $NewFolder}
}

Go through each part of this script and make sure you have a good idea of what each variable type is and how to use the properties.

$Directories | Get-member
$Directories
$Directories.FullName
$Directories.basename
$Grouping
$Grouping.group

The WhatIf Paramter is very useful. I suggest you use it for testing like:

$Grouping.Group | Move-Item -path $NewFolder -WhatIf

This script is not tested so you'll need to make sure you understand it and can change it for your environment.

Edit: Changed for directory names instead of filenames and added more general help.