For those who don't know, COBRA (Consolidated Omnibus Budget Reconciliation Act) is a law passed in 1985 [1] that compels companies to offer "continuing health insurance" to employees that leave the company for any reason other than "gross misconduct".
Basically it means that you can pay the company directly for your employer's health plan for a period of time, typically 18 months. The out-of-pocket will be more expensive, because your employer probably covered some of the premium cost. However, the premium as a whole is often cheaper than "individual coverage", since the company negotiates lower premiums with the insurance company.
The company will typically mail you the form to fill out. It's pretty short, basically just a "opt-in" box to continue getting health insurance through the company's plan.
One important detail: depending on your employer, health coverage probably ends on the last day of employment or the immediate end of the next month. However if something happens between you leaving and you filling out the form, COBRA still covers you because the law states that as long as you sign up within 60 days, it's "retroactive back to the event", the event being you leaving [2]. This covers the case of a catastrophe happening in the gap, e.g., getting hit by a car between the event and you mailing in the form.
Caveat: IANAL, so always check with a legal professional when evaluating these options. But COBRA is generally a useful thing and I typically recommend people take it unless they have another gig lined up already.
[1] http://en.wikipedia.org/wiki/Consolidated_Omnibus_Budget_Rec...