The SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS), a five-year spectroscopic survey of 10000deg^2^, achieved first light in late 2009. One of the key goals of BOSS is to measure the signature of baryon acoustic oscillations (BAOs) in the distribution of Ly{alpha} absorption from the spectra of a sample of ~150000 z>2.2 quasars. Along with measuring the angular diameter distance at z~2.5, BOSS will provide the first direct measurement of the expansion rate of the universe at z>2. One of the biggest challenges in achieving this goal is an efficient target selection algorithm for quasars in the redshift range 2.2<z<3.5, where their colors tend to overlap those of the far more numerous stars. During the first year of the BOSS survey, quasar target selection (QTS) methods were developed and tested to meet the requirement of delivering at least 15 quasars/deg^2^ in this redshift range, with a goal of 20 out of 40 targets/deg^2^ allocated to the quasar survey. To achieve these surface densities, the magnitude limit of the quasar targets was set at g<=22.0 or r<=21.85. We have defined a uniformly selected subsample of 20 targets/deg^2^, for which the selection efficiency is just over 50% (~10 z>2.20 quasars/deg^2^). This "CORE" subsample will be fixed for Years Two through Five of the survey. In this paper, we describe the evolution and implementation of the BOSS QTS algorithms during the first two years of BOSS operations (through 2011 July), in support of the science investigations based on these data, and we analyze the spectra obtained during the first year. During this year, 11263 new z>2.20 quasars were spectroscopically confirmed by BOSS, roughly double the number of previously known quasars with z>2.20. Our current algorithms select an average of 15 z>2.20 quasars/deg^2^ from 40 targets/deg^2^ using single-epoch SDSS imaging.