Pushing at birth in an upright or side-lying position is less likely to lead to an episiotomy and use of vacuum extraction or forceps compared with pushing and giving birth while lying flat or nearly flat on your back (supine position) or on your back with legs in stirrups (lithotomy position).
This stands to reason because when you are on your back, you are pushing 7 pounds or so of baby uphill against gravity. Also, pregnancy hormones relax the ligaments that normally keep the joints of your pelvis from moving. This allows your pelvis to open at the time of the birth to make more room for your baby to pass through it. Lying on your back, the weight of your body prevents the end of your spinal column (your sacrum) from flexing open.
For many decades, a common practice has been to coach women to bear down as long and hard as they can during contractions (sometimes called "purple pushing") once the opening of the uterus (cervix) is fully stretched (dilated). This came about in the belief that it was better for the baby for the pushing stage to be as short as possible and perhaps due to effects, in the 1940s and 1950s, when women were heavily drugged.
No research supports this practice, however. In fact, some studies that are beginning to look at this issue find that this style of pushing — as opposed to what women do when guided by their own pushing reflex — results in more injury and greater need for stitches without offering any benefits to the baby.
The pressure of the baby's head deep against your pelvic floor stimulates an innate pushing reflex ("urge to push"). When women follow their own inner sensations, they often don't begin pushing until some time after the cervix is fully opened. When they do bear down, pushes tend to last only a few seconds, and many women grunt or groan while they push. These differences may explain why you would be less likely to be injured or need stitches. Pushing in this manner allows the vaginal tissues to gently spread out around the baby's descending head. Pushing this way may also avoid overstretching pelvic floor ligaments and muscles. (Pelvic floor tips provides more ideas on how to reduce chances of a tear in your perineum.)
Women who have epidurals will usually be more limited in positions they can use. Many women who have epidurals may need caregiver-directed coaching as well because they have been numbed.
Specific concerns about mother or baby may also dictate pushing position. For example, squatting may be problematic in a woman with varicose veins because it can constrict blood flow from her legs, or the baby may do better with the mother pushing in one position versus another. Individual circumstances such as medical urgency to deliver the baby may also demand forceful pushing.
Pelvic floor resources offers books with more information on pushing position and technique at: http://maternitywise.org/mw/topics/pelvic-floor/options-pushing.html