Calling in Team Fortress 2 is communicating with your team to relay information so that everyone knows what's going on. Calling is one of the most important skills that a team can have, it's from calls that organized teamplays are made and it's also the backbone of the metagame. It's arguably the biggest difference between public games and organized competitive play.
This is the same reason lobbies are very often frowned upon and disregarded as any demonstration of skill, as the in-game voice system isn't the best solution and there isn't any obligation to use in-game voice chat or any type of external communication, often resulting in abnormal outcomes from risky plays that wouldn't happen if these plays were called.
Calls play an essential part in competitive play, so they should be treated as such and given special attention and work. Calling isn't often trained or practiced and most of the calling attributes a player has come from actual competitive experience, so it is natural for new players to feel overwhelmed by it or not being able to make proper calls. A call should be concise and clear, only with relevant information and no gibberish. It should also be spoken in a calm way that is understood by everyone, raging or shouting often only leads to a worse team atmosphere.
An example of a good (concise and clear) call would be:
"Scout on choke took 70."
An example of a bad or less clear call would be:
"Scout took a lot, why didn't he die?!?! Please kill him!"
The main caller is the backbone of any team, be it a Highlander or 6v6 team. It is the job of the main caller to communicate with the entire team, informing them when and how pushes, retreats and various other plays are to be carried out. It is essential that the main caller communicates clearly and concisely so actions are carried out as soon as possible. Teams which often fail to listen and react to the main caller generally fall apart due to a lack of focus on their enemies, positioning and the game as a whole.
Having a main caller is not a requirement nor obligation for every team, very often teams use different systems as they see fit to their playstyle, often using no maincaller at all or even 2 callers with strong maincalling presence.
A main caller should be a player that has a clear understanding of the metagame and a good gamesense, that knows how their team can take advantage of a certain situation or turn the match around with precise and concise calls. Having a strong presence in the team and being respected and heard by your teammates at all times is also necessary. A good microphone is recommended but not obligatory.
Main Calling in 6v6
The main caller is usually the Pocket Soldier, the Medic or the Demoman. This is because most 6v6 teams will focus their play around their combo to make use of über and general health advantages. That being said, a class that plays a big part in or around the combo is often recommended as a maincaller. However this is not true for all cases, as there are teams which have far more experienced players on flank positions that have good main calling abilities.
Main Calling in Highlander
Having a main caller in HL is much more important than on a 6v6 team, a player with a strong presence, that gives clear instructions is a plus, because HL is made up of 9 players in each team, the comms get usually a lot more noisy, often even resulting in a snowballing effect where calls are being talked over by other calls and there is no clear understanding of the direction the team should go. The main caller is usually the Heavy, Medic or Demo as these are the essential combo classes.
The in-game voice system is virtually unused in competitive play as it isn't perfect nor offers the amount of audio quality and customization the players need. So there are plenty of alternatives used, from TeamSpeak to Ventrillo, but the one that has become the staple of the competitive Team Fortress 2 community is Mumble, as it is very lightweight, open source, and also offers good audio quality and customization. Most teams will use this software to communicate and even the most active PUGs or Mixes channels/websites will enforce it, so it is an essential piece of software to the competitive scene. Another program that has recently started to be adopted by the competitive community is discord, which is also a very lightweight and in-browser app that allows for easy and convenient communication.