Share of search is the share of searches for your brand out of the total number of searches for all brands in the category. It's one of the few brand-building metrics a company can put together itself, almost for free, and its strength lies in being a leading indicator: a change in search share generally shows up before a change in market share, so it works as an early signal of both growth and problems. You build it by comparing the search volume trend for your company's name and that of your main competitors over the same period and territory. Watch out for two things: the names must be comparable and unambiguous, otherwise other meanings will get mixed into the figures, and track the metric as a trend over months, not weeks. Supplement it with the share of direct traffic, which confirms the same phenomenon from another angle.
See also: Brand equity, Incrementality, Entity Disambiguation.