Folks who are getting a divorce need life insurance for the same reason they needed it during the marriage: to protect a stream of income. This need is the same in an intact family as in a divorced one. The big difference is that married folks can ignore that need, but most divorcing folks cannot, as either their spouse's attorney or the court will make sure there is coverage. Of course if there isn't enough money to put food on the table there probably won't be money for life insurance either. In my experience, the courts recognize this and in such a case, may not require life insurance.
The amount of insurance needed is calculated by looking at what insurance the intact family had, what the education requirements, less savings, are likely to be, and what the support and health insurance costs will be for the payor. We then calculate the time value of money and the tax consequences in order to determine what a reasonable insurance replacement for the income stream will be. Also, it is perfectly fair to have the insurance reduced annually to reflect that the support has one year less to go.
The best explanation I have ever read on the types of life insurance available is from attorney/blogger Steve McDonough. Once you decide which type of insurance you are going to get, you may run into issues if you don't already have insurance. This article also appears to be a good resource on the subject.
Now you may be wondering what the picture of the dogs with the snapping turtle is doing here. It has NO relevance -- the dogs are my babies and I think it is a fun photo. You will be seeing more of them if my partners in crime allow.