Search engine marketing (SEM) is nothing but an internet marketing strategy to promote the website by increasing the visibility of the website in SERP. SEM is a broader phenomenon compared to SEO. SEO can be counted as a part of SEM, SEO uses an organic way of increasing the rank of website where as SEM uses both paid and unpaid methods to improve the ranking of website.
Many people get confused over the relation between SEO and SEM. Simply explained
SEO: Earning traffic with the help of free and unpaid listings
SEM: Buying the traffic through ads and paid listings along with SEO techniques.
SEM also can be explained partly with a mathematical expression like
SEM = SEO + PPC + Paid listings