I have noticed that participants in an HF exchange will often give number and a letter followed by a state, e.g. 1E Wisconsin. What does this mean?
So in a radio contest, there are definite rules about what information must be sent and received for the contact to be valid for contest points. The information that must be traded is known as the "exchange", in contesting lingo.
The example you gave sounds like the ARRL's Field Day. The ARRL don't describe Field Day as a contest; they describe it as "a local event and an opportunity for local amateur radio clubs to showcase the skills, science and technologies that make radio communication such a wonderful hobby and a valuable public service." However, in the course of this event, contacts are made according to rules. To many contesters, if it looks like an elephant, acts like an elephant, and smells like an elephant, then it's probably an elephant; in other words, many contesters view Field Day as a contest, and treat it as such.
According to the Field Day 2020 Rules (PDF), the exchange is the Field Day operating class and the ARRL or RAC section. In your example, "1E" is the operating class, which means a single-transmitter station running from a permanent location (i.e. home), but on emergency power. The ARRL section in your example is Wisconsin. Many ARRL sections have the same borders as US states, but not all; some states include several sections. Here's a web page that has several ARRL and RAC section maps.