A folk club is a pre-arranged regular event often weekly but can be fortnightly or monthly. They often alternate between a paid guest with supporting house/floor singers, and what is called a club night where the floor singers have a spot each and usually a visiting performer will be welcomed with a spot.
A singaround is similar to a club night but each performer usually gets one song at a time and the order of programme revolves around the room, with an MC in charge of encouraging performers.
A session, usually means a musicians' session and the lead is taken more or less at random with all joining in. Some are more suitable for learners and others for seasoned performers who know all of the standard session tunes. Most sessions allow for a newcomer to start a set of tunes.
All of these usually take place in pubs, sometimes in the bar, sometimes a private room.