@Robin's answer is good if the string is always a path (separated with "\"); in the general case, where the delimiter may be a different character, you can use
$string = "This-is-a-string-with-delimiters"
$lastword = ($string -split "-")[-1]
The -split operator defaults to splitting on a space, but will split on any character you choose to pass it, returning an array of strings, each string being the material before/between/after delimiters - in the example above, each word would be in a separate string in the array. Negative subscripts to an array count from the end of the array rather than the start, and are 1-origin rather than 0-origin - so $x[-1] is the last element of the array $x.
This technique also works on paths;
$path = "C:\Users\JSmith\Documents\Resume.doc"
$filename = ($path -split "\\")[-1]
will give you $filename -eq Resume.doc. Note that the delimiter passed to -split in this case is escaped, because the delimiter can be a regular expression, and the backslash ("\") is meaningful in regular expressions (it's the character that indicates that another meaningful character is to be "escaped", or its meaning ignored).
$name = $dir -replace '^.*\\',''