The Adventure

Where Are We Going?
Wargame, adventure game, role-playing game, White Box has been described variously and each depiction holds a degree of accuracy. Using miniature figures and measuring devices the game can play as a miniatures wargame. Explore the unknown and encounter challenges in the form of traps, puzzles and monsters and it is an adventure. Focus more on the words and actions of the individual characters and it's role-playing. My oldest group of gaming friends and I have always referred to the game sessions we play using White Box and its successors (regardless of focus) as "Adventures".
I think of the "Adventure" as what happens during play. It includes the preparation the referee does and the intention for what kind of dynamic the referee expects as a result. To my way of thinking the term "Adventure" covers all the various types of games that can be played using White Box and similar systems. The actual focus of play can vary widely and it is important that all at the table are working toward the same type of adventure.
One way to run an adventure is for the referee through verbal description to present the players with a situation and ask them what they want to do. Sometimes this is called a "sandbox" because the players can make just about anything out of the game they like. The referee follows their lead and improvises what isn't prepared ahead of time. As referee I find this method of play very entertaining.
Another obvious way to play is for the referee to develop a detailed storyline or plot and invite the players to role-play the characters in the story. This involves cooperation on the part of the players who may have to forgo some of their freedom of action in order to not take the story in an unexpected direction. It can be quite rewarding to play a central part in a good tale.
For some the goal of the game is for their characters to acquire treasure and experience, become rich and powerful and eventually join the ranks of the legitimate nobility. In this case the adventure may be anything that furthers this end. If treasure is found and experience earned the game is a success. It seems many early games consisting of a dungeon stocked with random monsters to kill and treasures to take were designed to do just this.
Exploration and discovery is another way to play the adventure game. A referee may spend a lot of time and creativity is developing a unique setting for the game and exploration of a detailed, original setting can be entertaining in itself. To discover a setting's secrets while perhaps making choices that impact the future of that setting can be great fun. One way to do this is the "hex crawl" which involves the party of adventurers traveling from one hex on a map to another uncovering whatever details the referee has to reveal in each location.
I was introduced to the investigation scenario by the game Call of Cthulhu and although I recall struggling at first to comprehend this new style of play, I credit CoC with teaching me to role-play a character. So much of the investigation adventure is asking questions of the NPCs that it is hard not to role-play. Like many others, I found that I liked this aspect of play and soon we were using it in all the adventure games played by our group.
A variation on the theme of character advancement is character development, or using the adventure as a means of playing out some personal aspect of the fictional character's personal growth. The character may develop and advance skills therefore becoming the hero they were meant to be, or they may resolve certain personal issues such as revenge, or experience growth such as forgiveness. They may pursue their passions or fall victim to their vices and this becomes the story that emerges during play. Certain games seem built around such in-game goals and The Burning Wheel instantly comes to mind.
I am a believer in pregame discussion - a session zero, in which the referee and players discuss what type of game they will be playing. Having shared expectations for the upcoming session or campaign can make play go much smoother and avoid tension at the table. It is a good idea to give some thought as to what kind of adventure you want to run as referee and make sure your players are on board for what you have in mind. Characters should be tailored to suit the agreed upon game style.