Chorus samples the input, slightly detunes it and mixes it with the original signal to produce a somewhat thicker, shimmering sound. Use it to thicken up background vocals, or to double the sound of brass and woodwind instruments. (source Behringer DeepMind 12 manual) The effect can be achieved by using multiple, slightly detuned oscillators for each note (source).
Flanger emulates the phase-shifting sound (comb-filtering) originally created by applying pressure against the flange of the reel on a tape recorder. This effect creates a unique “wobbly” sound that is quite dramatic when used on vocals and instruments. (source Behringer DeepMind 12 manual). From wikipedia: Flanging /ˈflændʒɪŋ/ is an audio effect produced by mixing two identical signals together, one signal delayed by a small and gradually changing period, usually smaller than 20 milliseconds. This produces a swept comb filter effect: peaks and notches are produced in the resulting frequency spectrum, related to each other in a linear harmonic series. Varying the time delay causes these to sweep up and down the frequency spectrum. A flanger is an effects unit that creates this effect.