In today's data-oriented world, just about every retailer has amassed a huge database of purchase transaction. Each transaction consists of a number of products that have been purchased together. A natural question that you could answer from this database is: What products are typically purchased together? This is called Market Basket Analysis (or Affinity Analysis). A closely related question is: Can we find relationships between certain products, which indicate the purchase of other products? For example, if someone purchases avocados and salsa, it's likely they'll purchase tortilla chips and limes as well. This is called association rule learning, a data mining technique used by retailers to improve product placement, marketing, and new product development.
R has an excellent suite of algorithms for market basket analysis in the arules package by Michael Hahsler and colleagues. It includes support for both the Apriori algorithm and the ECLAT (equivalence class transformation algorithm). You can find an in-depth description of both techniques (including several examples) in the Introduction to arules vignette. The slides below, by Yanchang Zhao provide a nice overview, and you can find further examples at RDataMining.com.