Biasedness is a property of a statistical method you're using to discover the 'true value' of a parameter (normally the relationship between two variables). If the method is 'biased', then it will not estimate the true value, except by luck. Typically, this property gets worse as you get more data: as your sample size grows to infinity, you become certain about the wrong number!
A common example would be if you're looking at the relationship between years of education and income. The typical estimate of the relationship between these two variables will be biased. This is because those with more years of education will tend to have better social groups, more supportive families, and other things that help one earn more independently of their education.
Consistency, a related property, is when your method gets better at predicting the true value as the sample size increases. Some methods may be biased but consistent, so that in small samples they tend to be wrong, but as the sample size grows, they zoom in on the true value.