The way a mother expresses milk typically depends on two factors: the reasons she is expressing milk and her comfort with a specific method. A mother who is separated from her baby to go to the movies, or who needs a little relief from breast fullness the first time her baby sleeps 6 hours straight, might find hand-expression a useful tool. A mother with a baby in the NICU should be using a hospital-grade rental pump to provide milk for her baby and maintain her supply.
Hand-expression is a technique in which a mother uses her hands to compress/squeeze milk out of her breasts. There are a number of good videos showing how it is done. It is almost always more efficient than a pump for expressing colostrum, which is available in such small quantities that the drops tend to get lost in a pump. For a mother who is very efficient, this technique could replace a pump during longer periods of separation from her baby; for most women, this technique is best used to make herself more comfortable or to express for short periods of time away from her baby. For more detailed visuals, please refer to the Stanford School of Medicine hand expression video available here.