See this great Stack Overflow answer.
I've also reproduced the whole thing below, and cleaned it up a bit.
I think that biases are almost always helpful. In effect, a bias value allows you to shift the activation function to the left or right, which may be critical for successful learning.
It might help to look at a simple example. Consider this 1-input, 1-output network that has no bias:
The output of the network is computed by multiplying the input $$x$$ by the weight ($$w_0$$) and passing the result through some kind of activation function (e.g. a sigmoid function.)
Here is the function that this network computes, for various values of $$w_0$$:
Changing the weight $$w_0$$ essentially changes the "steepness" of the sigmoid. That's useful, but what if you wanted the network to output 0 when $$x$$ is 2? Just changing the steepness of the sigmoid won't really work -- you want to be able to shift the entire curve to the right.
That's exactly what the bias allows you to do. If we add a bias to that network, like so:
...then the output of the network becomes sig$$(w_0x + w_1 \cdot 1.0)$$. Here is what the output of the network looks like for various values of $$w_1$$.
Having a weight of -5 for $$w_1$$ shifts the curve to the right, which allows us to have a network that outputs 0 when $$x$$ is 2.