How can I get powershell exception descriptions into a string?

How about:

$x = ($error[0] | out-string)

Is that what you wanted?

I took it a bit further because I didn't like the multilines from $error[0].InvocationInfo.PositionMessage.

Function FriendlyErrorString ($thisError) {
    [string] $Return = $thisError.Exception
    $Return += "`r`n"
    $Return += "At line:" + $thisError.InvocationInfo.ScriptLineNumber
    $Return += " char:" + $thisError.InvocationInfo.OffsetInLine
    $Return += " For: " + $thisError.InvocationInfo.Line
    Return $Return
}

[string] $ErrorString = FriendlyErrorString $Error[0]
$ErrorString

You can look at what else is availible to construct your own String via:

$Error | Get-Member
$Error[0].InvocationInfo | Get-Member

If you want a bit shorter message (more user friendly sometimes?) than @tomasr suggests this will do:

$error[0].ToString() + $error[0].InvocationInfo.PositionMessage

You will get something like:

Cannot find path 'C:\TEMP\_100804_135716\missing' because it does not exist.
At C:\TEMP\_100804_135716\test.ps1:5 char:15
+   Get-ChildItem <<<<  missing

This technical info will be excluded:

+ CategoryInfo          : ObjectNotFound: (C:\TEMP\_100804_135716\missing:String) [Get-ChildItem], ItemNotFoundException
+ FullyQualifiedErrorId : PathNotFound,Microsoft.PowerShell.Commands.GetChildItemCommand