What is an Infobox?
"An infobox is a panel, usually in the top right of an article, next to the lead section (in the desktop view), or at the very top of an article (in mobile view), that summarizes key features of the page's subject. Infoboxes may also include an image, and/ or a map. Infoboxes generally use the template software feature. (It is possible to hand-code an infobox using table markup, but this loses the advantages of standardization and reusability.) The templates have parameters; to work properly, the parameter values have to be specified when the template is inserted in the page. This allows each infobox to show information relevant to the article subject, while requiring only a minimal amount of coding within each article."
Wikipedia, Manual of Style/Infoboxes
Infoboxes are generally designed to be used across a common class of article, usually related to a Category.
Why do we Infobox?
Two basic reasons:
1 - Information Presentation - To quickly present a summary of the article to allow a reader to be presented with the most relevant facts quickly and consistently.
2 - Information Structure - Tools are available to use the Infobox data in automated ways, not the least of which is Dynamic Page Lists aka DPL.
Design of an Infobox Template
The two "why"s provide different design criteria, both should be considered.
1 - Data needed to quickly present the essence of the article to the reader across the type of article the infobox is being designed for. Not so much though that it is overwhelming, as that defeats the purpose.
2 - Data needed to summarize, classify and sort the Dynamic Page Listings used for indexing this class of article.
Using an Infobox Template in your article
Fairly straightforward, check the individual template's documentation. View examples of other uses elsewhere.
Linking in Infoboxes