I'm not sure what 'modes' are, but you can look here:
http://en.wikipedia.org/wiki/Mode
Degrees of freedom (DOF) are the number of dimensions in which something can move. For example, there can be up to three translational DOFs for most objects; x, y and z. There are also rotational DOFs; one about each of the three axes for objects which are not rotationally symmetric about that axis, and vibrational DOF if there are chemical bonds which can stretch and compress like springs...