I would say that jumping rope is definitely the best way to go, as if done right it is actually a lower impact exercise. Mainly to correctly jump rope, jump using the ball of your foot and keep your upper arms close to your body while the lower arms are directed out from your body (though not at 90 degrees), and keep the main movement in your wrists.
Running or jogging puts a large amount of stress and negative impact (over long periods of time) on joints in comparison to jumping rope, this is due to the momentum involved in running (which magnifies the weight of the person) and the fact that the entirety of this force is taken on by one leg, instead of both (as in jumping rope).
Also, jumping rope increases endurance, rhythm, reaction speed, co-ordination and balance more effectively than jogging or running. The endurance is because jumping rope accesses and uses your core muscle groups (as well as others) at a much quicker and intense rate than running or jogging can. The rhythm is due to the the repetition of the exercise, this is heavily connected with reaction time. Reaction time is increased in reaction to the rhythm, the brain learns to quickly calculate the speed of the rotation, which in turn leads to anticipating the next jump, this also helps to increase brain activity in relation to memory. Co-ordination is increased due to the combination of jumping and moving of the arms, this also helps to tone the upper body much more successfully than jogging or running, as both sides of the brain are used while jumping rope it increases spacial awareness (which has a large impact on reaction time, balace and co-ordination) and body awareness (also has a large impact on reaction time, balance and co-ordination). Balance is increased as noted in the previous sentence, however it is also due to the strengthening of the core muscles.
Also jumping rope definitely combats depression, as any type of physical exercise has a positive repercussion against depression. Depression can be caused by many things, high up on the list however are: lack of physical fitness (which inhibits self-confidence, this can be combatted through any physical exercise, jumping rope is a very portable example), lack of vitamin D (which is from a lack of sunlight, this can be combatted (via jumping rope) by going outside to exercise), and lack of self-confidence (which can be improved through improving your image, however to be confident you have to be happy with yourself, a way to do this is to act as though you are watching yourself from the outside, another way is to react to situations in a way that you believe to be admirable and moral, surround yourself with quality (not necessarily quantity) people (these people should be supportive, friendly, intelligent enough to provide mental stimulus but perhaps not so intelligent to make you feel overly stupid, and honest with you. Avoid negative, abusive, aggressive, or depressing company).
I hope this helped out! also here is a site that shows 23 different types of jumping rope: http://www.bloomtofit.com/23-skipping-rope-exercises-extended-video