Imagine you have $1000 in the bank … if you are spending $100 month you will have nothing in 10 months but if you decide to take $50 each month it will last 20 months. Which would you choose? Among other things, it would depend upon whether $50 was enough to do what you need to do.
A carbon budget is similar. A carbon budget is a limit on the total amount of CO2 in the atmosphere. We can emit CO2 at a constant rate and we will hit the limit fast, but if we reduce our annual emissions gradually we will have longer before we hit the limit.
There are a variety of carbon budgets and this document by Carbon Tracker discusses the differences (pdf).
As far as individual countries are concerned their carbon budgets depend upon how the global carbon budget is shared: on a per-capita or some other basis such as GDP.
The Intergovernmental Panel on Climate Change (IPCC) reports specify different carbon budgets for a number of risk levels but the one we are using is "likely" which they define as greater than 66% (2 in 3).
The following chart was extracted from the recent The Sky's Limit report (pdf) by Oil Change International. It illustrates how much the developed reserves (those expected to be recovered from existing wells including reserves behind pipe) exceed the carbon budgets for a likely (> 66%) chance of global temperatures increases staying below 2 degrees and only a medium (> 50%) chance of remaining below 1.5 degrees.
Sources: Rystad Energy, IEA, World Energy Council and IPCC.