SEO: Search Engine Optimization
In layman's terms, improving your rank in search engine result listings on Google, Yahoo, Dogpile, etc. Often you will hear the terms "organic" or "natural" search results which refer to the areas of the results pages that are not "sponsored" or paid for advertising. Actions to improve SEO include keyword optimization, relevant content creation, creating directory listings, and much more.
SEM: Search Engine Marketing
The easiest distinction between SEO and SEM is if that the place the link shown is paid for, it's SEM. SEM refers to ads and links that are "sponsored" to be placed in designated sections of results pages. If you are participating in paid advertising on a search engine, your ad may show within the special sections of the page reserved for advertisements. Another major distinction, when you "turn off" your SEM campaign, your ads will stop showing where as SEO content is more permanent.