When a lawyer tells me that he would take my case on contingency basis, what is that supposed to mean?

It means he figures you have a pretty good case and there is enough money involved for him to pursue the case without any money up front. He will not receive any compensation unless he wins. Then he gets at least a percentage of the judgment or settlement which ranges from 1/4 to 1/2 with the average being 1/3. However, some lawyers add a plus expenses which can cut into your share of the money.