Next To
Being Next To someone means you are close enough to touch each other physically. In other game systems, this means you are within reach. Since this system has no concept of reach, you have to be able to touch. In miniatures, it means your bases are touching. For the sake of not being stupid, if you are all engaged with the same single opponent, you are all Next To each other (DM might fiat on particularly big monsters like Huge ones).
Engaged
Engaged is very similar to Next To, except for one important distinction - you are actually in combat against each other at melee ranged. Thus, you can be Next To an enemy and not Engaged because you aren’t fighting each other (see Free). However, you cannot be Engaged with someone and not be Next To. Other systems may refer to this as threatening each other.
Free
Free is the opposite of Engaged. You cannot be Free and Engaged. You can be Next To someone and Free of them, as is the case with your allies most of the time. If you are not Engaged with anyone, you are by default Free.
Nearby
Nearby something means you can get to it with a single move action, assuming conditions do not make it difficult. Creatures you are Next To or Engaged are, by definition, Nearby you. Nearby is a range abstraction that is up to roughly 30 feet to 100 feet if you really need to measure it out.
Far Away
Far Away from something means you are two move actions away from being Next To under normal conditions. You cannot be Far Away and Next To or Nearby the same thing without some kind of spatial warping or special circumstances.
Disengaging
Disengaging is the act of trying to stop being Engaged without drawing an opportunity attack. For most folks without special training, this involves quite a bit of juking and dodging. Thus, Disengaging requires a move action. Additionally, you must roll a Normal save (11+), adding one to the roll for each person past the first you are trying to Disengage from. Fortunately, if you succeed, you are considered to have “popped” Free, and can move away without drawing any opportunity attacks with the same move action. If you fail, however, you have used your move action juking and dodging and were unable to Disengage.
You always have the option of not attempting to dodge and just accepting the opportunity attack.