A Citation is a mention of your business details online and does not necessarily include a link to your website (a backlink).
One of the biggest factors in your ability to rank well in Local Search is the number and quality of citations for your business.
You should aim to build as many citations for your business as possible with a focus on building those citations on higher quality services first. Each of these citations builds Authority around your business and tells the search engines that you are a real business in your area with multiple indications you are who you say you are.
Your presences are a subset of your citations as it is possible to have citations that appear in places that do not directly relate to your business such as Blog Posts, Articles and Job Advertisements.
It is not enough to just build citations for your business, you also need to make sure those citations are as accurate and consistent as possible across all of your citations. Without this consistency your citations may not be considered when determining your business's rank in local search as the search engine cannot be sure that citation is for your business or not.
As an example, Search Engines are completely automated algorithms that generate a score or rank for your business in local search. When they discover a citation for a business they need to decide which business that citation is for and will analyse your business information to decide. If your Name, Address or Phone Number differs from citation to citation it is likely the algorithm will consider some of them to be for separate businesses leading to a lower potential rank for your business.
This could be the difference between you appearing on page 1 for your local terms or not.