Side breakfall is the most important breakfalling technique, since the vast majority of reaps, sweeps, trips, throws, takedowns, and walks across icy parking lots end with landing on your side. It may seem like landing on your side will be a harder impact, since the side has a smaller area than the back -- but you will also land on the side of your leg, so there is no net reduction of surface area.
Level 1 - Squatting side breakfall
Squat down into a little ball, standing on the balls of the feet, as you did when learning how to do rear breakfalls. Cross your arms in front of your chest. Tuck the chin -- we cannot stress that enough. Then, extend one leg straight forward, as though your were doing a traditional Russian folk dance. This will feel awkward, and one may feel like you will topple over, but that's okay, falling is the next step!
Roll towards the extended leg. Keep the other leg bent, with its foot flat against the mat. Upon landing, forcefully exhale with nogare breathing. Slap the mat with the hand closest to the ground; the arm lands last, and remains straight, and at a 45° angle from the body.
Practice this technique on both sides until you feel comfortable with it.
[video of squatting rear breakfall, from the front and from the side, fast and slow]
Level 2 - Side breakfall
Starting from a ready stance, facing #1, exercise kick to wither #2 or #8, and cross the same-side arm over your chest. Then, lean to the side until you topple over, landing in a side breakfall, as described above.
[video of side breakfall, from the front and from the side, fast and slow]