Google+ Custom URLs means Facebook-Twitter type username post fix after the domain name.
Currently Google Plus profiles are identified by a number string, other than this Google is planning for Google Plus custom URLs to make profile addresses more easy.
Sam Laird in one of his post says that:
“The Google+ team began notifying users Thursday morning of the option to claim a pre-approved custom URL or apply for a different one.”
Company on its Policies & Principles page updated that customs URLs are under development and this program may change. They reserve the right to reclaim custom URLs or remove them for any reason, and without notice. Right now custom URLs are free but they may start charging a fee. However you will be getting notified before so that you can stop your participation for custom URLs.