SEO or organic/natural search engine optimization is a subset of SEM, where the focus (of SEO) is in keyword research, competitive analysis, on-page optimization and off-site optimization to improve visibility and the resulting (targeted) traffic of a web site on GYM (Google, Yahoo, MSN).
SEM on the other hand is the containing category, which includes paid search (pay per clip, pay per post, cost per click or pay per click), social media optimization/marketing, viral marketing and other flavors of marketing, that involve search engines, but drive traffic through related (yet diverse methods). The approaches used in these other marketing methods are refined and revised circumstantially and situationally or as search algorithms change.
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