Changing file permissions with PowerShell is not to difficult but not as straight forward as you would think. Listed below are how to do some common activities.

Adding permissions to an object.

$path = "c:\myfolder" #Replace with whatever file you want to do this to.
$user = "mydomain\myuseraccount" #User account to grant permisions too.
$Rights = "Read, ReadAndExecute, ListDirectory" #Comma seperated list.
$InheritSettings = "Containerinherit, ObjectInherit" #Controls how permissions are inherited by children
$PropogationSettings = "None" #Usually set to none but can setup rules that only apply to children.
$RuleType = "Allow" #Allow or Deny.

$acl = Get-Acl $path
$perm = $user, $Rights, $InheritSettings, $PropogationSettings, $RuleType
$rule = New-Object -TypeName System.Security.AccessControl.FileSystemAccessRule -ArgumentList $perm
$acl.SetAccessRule($rule)
$acl | Set-Acl -Path $path

With the above code you can set the rights on a folder. You could set the $InheritSettings to none if you want to apply it to a file instead.

Valid settings for Rights are as follows:

Setting Description
AppendData Specifies the right to append data to the end of a file.
ChangePermissions Specifies the right to change the security and audit rules associated with a file or folder.
CreateDirectories Specifies the right to create a folder.
CreateFiles Specifies the right to create a file.
Delete Specifies the right to delete a folder or file.
DeleteSubdirectoriesAndFiles Specifies the right to delete a folder and any files contained within that folder.
ExecuteFile Specifies the right to run an application file.
FullControl Specifies the right to exert full control over a folder or file, and to modify access control and audit rules. This value represents the right to do anything with a file and is the combination of all rights in this enumeration.
ListDirectory Specifies the right to read the contents of a directory.
Modify Specifies the right to read, write, list folder contents, delete folders and files, and run application files. This right includes the ReadAndExecute right, the Write right, and the Delete right.
Read Specifies the right to open and copy folders or files as read-only. This right includes the ReadData right, ReadExtendedAttributes right, ReadAttributes right, and ReadPermissions right.
ReadAndExecute Specifies the right to open and copy folders or files as read-only, and to run application files. This right includes the Read right and the ExecuteFile right.
ReadAttributes Specifies the right to open and copy file system attributes from a folder or file. For example, this value specifies the right to view the file creation or modified date. This does not include the right to read data, extended file system attributes, or access and audit rules.
ReadData Specifies the right to open and copy a file or folder. This does not include the right to read file system attributes, extended file system attributes, or access and audit rules.
ReadExtendedAttributes Specifies the right to open and copy extended file system attributes from a folder or file. For example, this value specifies the right to view author and content information. This does not include the right to read data, file system attributes, or access and audit rules.
ReadPermissions Specifies the right to open and copy access and audit rules from a folder or file. This does not include the right to read data, file system attributes, and extended file system attributes.
Synchronize Specifies whether the application can wait for a file handle to synchronize with the completion of an I/O operation.
TakeOwnership Specifies the right to change the owner of a folder or file. Note that owners of a resource have full access to that resource.
Traverse Specifies the right to list the contents of a folder and to run applications contained within that folder.
Write Specifies the right to create folders and files, and to add or remove data from files. This right includes the WriteData right, AppendData right, WriteExtendedAttributes right, and WriteAttributes right.
WriteAttributes Specifies the right to open and write file system attributes to a folder or file. This does not include the ability to write data, extended attributes, or access and audit rules.
WriteData Specifies the right to open and write to a file or folder. This does not include the right to open and write file system attributes, extended file system attributes, or access and audit rules.
WriteExtendedAttributes Specifies the right to open and write extended file system attributes to a folder or file. This does not include the ability to write data, attributes, or access and audit rules.

Valid Inherit settings:

Setting Description
ContainerInherit The ACE is inherited by child container objects.
None The ACE is not inherited by child objects.
ObjectInherit The ACE is inherited by child leaf objects.

Valid Propagation Settings:

Setting Description
InheritOnly Specifies that the ACE is propagated only to child objects. This includes both container and leaf child objects.
None Specifies that no inheritance flags are set.
NoPropagateInherit Specifies that the ACE is not propagated to child objects.

Removing permissions from an object

$path = "c:\myfolder" #Replace with whatever file you want to do this to.
$acl = Get-Acl $path
$rules = $acl.Access | where IsInherited -eq $false #Gets all non inherited rules.

#Filter $rules to however you want to decide on which rule you want to remove.

#To target a specific user or group as an example.
$targetrule = $rules | where IdentityReference -eq "MyDomain\MyUserAccount" #Leave domain off for local accounts.

$acl.RemoveAccessRule($targetrule)
$acl | Set-Acl -Path $path

The above script will strip rules off a file or folder. Make note of the IsInherited property on rules.