LSI is a term that now has multiple meanings. It is related to the term "Latent Semantic Analysis" (LSA), but refers to the practice of using LSA for indexing text. From an academic standpoint, LSA is a set of methodologies for analyzing the relationships of terms used in written text. For more see http://en.wikipedia.org/wiki/Latent_semantic_analysis
For SEO, LSI typically refers to the analysis a search engine might do to analyze text as part of the ranking algorithms. No one really knows how search engines really use LSI (and it is probably a moving target anyway), but it is pretty certain that LSI is used in various ways to determine quality, relevance and other factors. For example, a search engine might use LSI to determine if one set of text is contextually relevant to another when looking at links between pages, or it may use LSI to determine if a given block of text is real natural language or artificially generated.
You can find endless discussions on the details of how LSI might be used by the search engines, but at the end of the day, focus on quality and relevancy, and you will be OK. Ultimately that is what the algorithms using LSI will be looking for.