The Phaser is the first of the two effects to be invented (around the '60), and consists into splitting the signal in 2 identical ones, and into applying to one of them a filter that alters its phase. By summing the two signals; the obtained effect it's a rhythmic phase cancellation that variates according to the settings, thus creating the "oscillating" sound typical of phaser.
The Phaser, unlike the flanger, doesn't work by delaying the second signal and summing it to the first one, therefore the sound is less aggressive.
Additionally, the output can be fed back to the input for a more intense phase shifting, creating a resonant effect by emphasizing frequencies between the wave notches.
The Flanger effect is produced by summing two identical signals together, with one signal delayed of a small and gradually changing period, up to some centisecond. This produces a phase cancelling effect that variates according to the variating lenght of the delayed signal, producing an effect called "Comb Filtering", named after the visual effect this process does to the waveform.
Variating the time causes the two tracks to sweep up and down the frequency spectrum, and the flanger is the effect dedicated to produce this result.
Part of the output signal is sometimes fed back to the input, producing a resonance (sometimes even inverting the phase of the fed back signal) which further enhances the intensity of the effect.
One of the most famous guitar players that has used this effect on many songs is Edward Van Halen.
Wok Flanger - a stompbox style flanger
Kjaerhus Audio Classic Flanger - simple to use
Kjaerhus Audio Classic Phaser - simple and essential.