Your busy life pulls you in all sorts of directions, so it is inevitable that you will have to cancel an appointment from time to time. It can be dicey, given that anything that requires an appointment usually has some degree of officialness to it. The art of cancelling an appointment hinges on striking a balance between feasibility and seriousness with your reason.
While you don't want to promote bad karma, citing a family illness is one way to get out of an appointment that will elicit little questioning from people. Be sure that the person who is allegedly sick isn't in contact with the person you're cancelling with. For example, cancelling an appointment due to a sick parent with a doctor that is your entire family's physician would not be wise (see Reference 1).
This is a tad vague, but much like with the family illness excuse, it will garner little questioning from the person with whom you are cancelling. It helps to attach a category to the personal reason, such as illness or emotional strife. Make it clear that there is a pressing, unavoidable issue that you have no choice but to deal with. You can do this by sounding distressed or sad, but be careful not to oversell it (see Reference 2).
If the appointment in question will eventually entail payment for services rendered, unforeseen financial difficulties provide you with an excuse that is essentially open and shut since you either can pay or you can't. If the appointment is not directly related to money, you can still play the financial card by saying that you don't have the gas money or that your car is not drivable due to unaffordable, costly repairs.
If you're like most people your job is your livelihood, therefore unexpected and time-consuming events related to it will rarely be questioned. One tried-and-true excuse that works with virtually any type of occupation is saying that someone else is sick, but the job still has to get done and you're the person responsible. Feel free to get creative, but make sure that there aren't any holes in your story that could expose you, such as the person you are cancelling with knowing that you may in fact be between jobs.