When your Google Plus account is setup, you are given an assigned URL or 'user ID' that looks something like this: plus.google.com/129876534098l
This long string of numbers after the '.com/' is representative of your username and by changing this part of your URL from a long string of numbers to your name for instance, you make it much easier for others to find, follow and remember who you are on and off Google Plus.
A bit on Custom URLs
There are a few different URL patterns out there, which vary depending on the social platform. For instance, Twitter URL's will always look like: twitter.com/username, or LinkedIn URL's appear as: www.linkedin.com/in/username whereas Google+ URLs appear as: plus.google.com/+username (or user ID).
The idea behind the custom URL or vanity URL, is that it makes it easier for search engines and people to find you online. It's also an arguably good way to generate some SEO behind your name.
Selecting a Username
There is the possibility that someone else (with the same name) may have registered your desired username; because usernames need to be unique, you may find that you have to get creative in making an alternative username.
Depending on the social platform, users may only give your profile a few seconds before deciding whether to follow you or not. Given this tiny window of time, it is best to optimize everything you can about your profile's 'first impression' - what might people remember about your profile after a quick glance?
A customized URL is another feature you can take advantage of to increase your searchability on Google Plus, give your name some increased SEO and is a good way to fine-tune your profile's overall first impression. I hope this blog post was helpful and that whoever is reading this is having a great day!