When someone says “Pubs” or “pub games” in the context of gaming, they are actually referring to the term “Public Games”. Before online competitive gaming and the matchmaking system became very popular, the term “Public Games” or “Pubs” was used to refer to LAN or online lobbies created for anyone to join. In other words, a Pub game was a LAN match or an online lobby which included not only people who know each other but people who were complete strangers to each other as well. Meaning that anyone can join them.
These days, with the popularity of online gaming, there are tons of public lobbies that anyone can join. Take the lobby section for Battlefield 4 for example. Anyone can join whichever server they want from the server browser, as long as they have Battlefield 4 installed on their computer. Not only that but all matchmaking systems for online videogames will send you into a public game or Pubs. There is no restriction for who can use the matchmaking system and play the game. If you are not playing with your friends, you will be paired with complete strangers.
Where the term Pubs is useful is with the competitive gaming community that regularly conducts small-scale tournaments and matches. This requires them to create lobbies that only specific players can join, which are the ones taking part in the tournament or match. So they often use the term “Pubs” to refer to matches other than organized matches or tournaments.