CLC stands for Certified Lactation Counselor. Some lactation consultants started off as CLCs just because they took the 40 hour CLC course. The CLC course is one of the many ways a lactation consultant candidate can meet part of the necessary educational requirements to sit for the boards. I took this course on the Cape in Massachusetts in 1998.
Some CLCs have experience working with breastfeeding moms and babies, but it is not a requirement.
IBCLCs (International Board Certified Lactation Consultants) use the name lactation consultant, though anybody can use this title, as it is not trademarked. It is important to find out if your lactation consultant is a board certified lactation consultant. You will also want to know her level of expertise and how she keeps her skills up to date.