"Degrees of freedom" refers to the number of free choices we have of number to specify a given situation. If a problem refers to the motion of a particle that could be anywhere on a given line then the particle has one degree of freedom since we could choose an arbitrary point on that line to be "0" and specify the position of the particle as the distance, positive or negative from that point. If the particle can be anywhere on a given plane it has "two degrees of freedom" since we could designate its position on an xy coordinate system. If the particle could be anywhere in space, it has three "degrees of freedom". Extending that, if the particle could be anywhere in space with any velocity vector, then we need 6 numbers to specify it three coordinate for its position and three for its velocity vector 6 "degrees of freedom".
