LSI Keywords are essentially keywords related to the keyword that you search for on search engines such as Google. In short, they are keywords that are related to your main term. They help support your content and give it more context for both the search engines and your users.
Latent semantic indexing (LSI) keywords are simply keywords related to the topic your page is about, and with Google's Panda update, they are looking for more of a topic match instead of a keyword match. How they do this is by looking for LSI keywords. The idea is that you would just naturally include them as you write. So by simply writing naturally, you will have found them.
Say for example you have a traveling site. The idea is that you would naturally include related words like "flights", "accommodation", "visit", "great place for a vacation" in the content. These are LSI keywords for your "traveling" topic.
Semantic search is a data searching technique in a which a search query aims to not only find keywords, but to determine the intent and contextual meaning of the words a person is using for search.
And here's where things get a little bit fun with LSI keywords: LSI keywords change all the time based on the current trend.
"Oculus Rift" is definitely not semantically related with "Virtual Reality" 5 years ago; but fast forward to today, Oculus Rift is basically a synonym of virtual reality.
There are many ways you can keep track of LSI Keywords, but here's what I would recommend: LSIGraph: LSI Keyword Generator.