Circumcise literally means "to cut around." In this context, this almost always means the removal or cutting of the male foreskin.
In terms of the most common usage, the removal of the foreskin is often done when the male is very young, but is occasionally done as an elective procedure as an adult (BME focuses exclusively on adult and elective circumcisions as topical, and the "official" stance of BMEzine is that routine infant circumcision is not supported due to lack of consent). There are some medical conditions that make circumcision potentially desirable. Other people desire circumcision for primarily aesthetic reasons, either for the uncovered glans, or for the look and feel of tight skin over the shaft of the penis.
If you want a circumcision, doctors are the obvious choice. It is generally a no-questions-asked procedure. There are also a variety of private non-medical practitioners (cutters) willing to do this procedure. There are some tools such as Tara KLamps which can in theory be used at home.
Outside of infection and bleeding, there is some risk of loss of sensation due to the thickening of the skin of the glans. Some people lose some degree of ejaculatory control due to loss of fine nerves. However, almost everyone is pleased with the overall result.