What is the effect of bandwidth on a digital signals?

Bandwidth is an inherent characteristic of a given transmission channel, or is determined by the narrowest-bandwidth component of the system. The bandwidth of a channel will limit the possible attainable data rates. This is shown simply by Shannon's Theorem, which states

C = B log_2 (1+P_signal/P_noise),

where C is the channel capacity in bps, B is the channel bandwidth in Hz, P_signal is the power of the detected signal in W, and P_noise is the noise power of the detected signal in W.

As an example, consider a standard phone line (i.e., using a dial-up modem). Standard phone lines have a bandwidth of about 3.4 kHz and a signal-to-noise ratio of about 10,000. Using this information, we get

C = (3.4 kHz) log_2 (1+10,000) = 45 kbps.

Dial-up modems can actually get as high as 56 kbps, but that is beyond the scope of this question. In general Shannon's Theorem can provide a fairly accurate way to predict the possible data rates for a given transmission channel if the bandwidth and resulting signal and noise powers are known.