So, take your classic example of earning wages. If you get paid $10 per hour, you could work 8 hours and make $80, or you could work 2 jobs, and get paid $160 for 16 hours of work. If you work the same job for 16 hours, you can earn an extra $40 for overtime, meaning that you can earn $200. If you create a graph for the first example, plotting the goods purchased by your income on one axis, and the amount of leisure that you have, then you end up with a nice smooth line showing all possible outcomes that can occur for your labor. This is a BUDGET CONSTRAINT. (Stick with me... this is going somewhere fun...)
When you plot out the second situation, you have a line with one slope for the first 8 hours, and a second slope for the second 8 hours. Your budget constraint has a kink in it. Economists have called this a KINKED BUDGET CONSTRAINT, because they tend to use terms that look easy to understand. Generally, you see these kinked budget constraints when government policies are in play, like time and a half for overtime, or welfare payments, or the earned income tax credit.
One thing that I hate about the study of economics is that it takes concepts that sound intuitive, then give them terms that sound really, really, really, really bad if you see them over and over again. Take kinked budget constraints. Read this a couple dozen times when you are studying for an exam, and your brain will probably go here at some point:
tee hee hee... kinky budget constraints... hee hee...