A connection is made whenever a client (i.e. a browser or a mobile app) requests a resource from a server (i.e. a web page, CSS, JS, image, etc). From a servers point of view "concurrent connections" is the count of the number of clients that are connected at the same time.
In a traditional web page request, your browser will open a connection to the server, request and receive the HTML, close the connection, parse the HTML and then request the JS, CSS and Images. Browsers are typically able to open multiple connections to a server and request these resources in parallel, and then close those connections. Most are able to reuse a connection to request additional resources, but once all resources are downloaded the connections are closed. These connections are typically only open for a few seconds at most.
Web sockets impact this because they create a persistent connection between the client and the server while the client is running. For instance, if your server supports a maximum of 100 users running your app, all users will have at least one connection to your server open. If your server only supports 100 concurrent connections then you won't be able serve any other users, including serving the initial HTML.