To complement Keith Hill's helpful answer with additional information:
Both "$computer\admin" and the unquoted form, $computer\admin, would work, because unquoted string arguments are implicitly treated as if they were "..."-enclosed (double-quoted), i.e. as expandable strings that perform string interpolation (replace embedded variable references and expressions with their values), as opposed to verbatim strings ('...', single-quoted) that do not interpret their content.
When in doubt, use "..." explicitly, notably when the string contains metacharacters such as | and <
For the complete rules on how unquoted tokens are parsed as command arguments, see this answer.
Pitfalls:
The partial quoting you attempted:
'$computer'\admin
even if corrected to "$computer"\admin to make interpolation work, would not work, because PowerShell - perhaps surprisingly - then passes the value of $computer and verbatim string \admin as two arguments. Only if a compound string with partial quoting starts with an unquoted string is it recognized as a single argument (e.g. $computer"\admin" would work) - see this answer for more information.
Another notable pitfall is that only stand-alone variable references such as $computer and $env:COMPUTERNAME can be embedded as-is in "..."; to embed an expression - which includes property access and indexed access - or a command, you need to enclose them in $(...), the subexpression operator. E.g., to embed the value of expression $someArray[0] or $someObj.someProp in an expandable string, you must use "$($someArray[0])" or "$($someObj.someProp)" - see this answer for the complete rules.