Peer Review is a process articles in scholarly journals go through. Experts on the topic the article is written about read and comment on it, and only the best articles are accepted and make it into the journal. Peer reviewed articles must be of high quality, and they always add new knowlege or information about their topic to the sum of human knowledge.
Indexing means creating a searchable list of many different books, journal articles, or other printed or electronic sources (it usually means journals, though). This lets a searcher find out what has been published about a particular topic. Indexes do not always have a direct link to the full article or source, they're mostly for finding out what exists out there.
Databases are large collections of journal articles, grouped in some way, usually by topic. Many databases have links to the full text of articles, although some mostly do indexing. Databases are the best place to find scholarly articles or research.
