Your parameter estimates are biased. They are systematically either too large or too small.
Let's say the true parameter is $\theta$, and your estimate is $\hat \theta$. The bias is, as you say, the expected difference between the two, $\mathbb E[\hat \theta - \theta] = \mathbb E[\hat \theta] - \theta$ (true parameter treated as a constant). The bias is decided by the expected value of the estimated parameter. Downward bias means the parameter estimates are expected to be smaller than the true value they estimate; upward bias means the parameter estimates are expected to be larger than the true value they estimate.
It is not always possible to tell what the bias is as you need to know the true value $\theta$, which in many cases you can't.