The "burn rate" is the speed at which a company is spending their free cash less any incoming revenue. A burn rate is often specified hand-in-hand with the amount of runway that a company has; the amount of time a company can continue spending at their given burn rate before running out of operating funds.
A good (or bad) example of a burn rate is Über. The car ride company has been said to have a burn rate of up to $2 billion a quarter. This means that, despite every dollar of revenue they bring in in a quarter, the company "burns" through $2 billion of their cash on hand every quarter.
Typically, for any small business, a burn rate is a bad thing. The idea is to get to at least break even as quick as possible and self-finance any further expansion. This is because the money must come from somewhere, either from the entrepreneur's own pockets, through friends and family or from accredited investors and no one likes to see the money they've worked for "burned" so that a company can reach an often illusive goal.