First, the history. Data integration is the name the vendors have adopted to replace the ETL (Extract Translate Load), data cleansing, and data warehousing tools of days gone by.
These tools actually pre-date the notion of EAI, and were really the first sets of technology designed to deal with data and the use of that data for decision support (business intelligence now). They would extract large amount of data from a single or several operational data stores, clean the data, roll up the data, and put it in another data store, typically the data mart or data warehouse for analysis. From there somebody would "mine" the data to extract relevant information, such as productivity over time, sales effectiveness over years, profit by division, you get the idea. Very powerful notion for its time, and very powerful technology today.