As the analog filtering, digital filtering is a method to alter or remove unwanted frequencies components from a signal. (I guess you know about lowpass, highpass, bandpass and bandstop filters).
Generally, a filter has a form of:

y[n] = 1/a0*(B x X –A x Y)
where y[n] is the output response
      B is a k-dimension vector – contains forward coefficients;
      X is a k-dimension vector – containing past k samples from the input signal;
      A is a (n-1)-dimension vector – reverse coeff.
      Y is a (n-1)-dimension vector – contains past n-1 outputs from the filter.

In most filter designs coefficient a0 is =1 (anyway it can be containd in B vector).

Now, if A vector contains just null elements the filter will be an finit impuls response filter (FIR) and if not the filter is called infinit impulse response (IIR- the output depends of the past outputs and is assumed that it’s response will be infinit) .
So, you can see – the implementation is not such a big problem – you must multiply some vectors. A simple filter is a moving average, that is al the elements from the B vector have the same value =SAMPLE_COUNT.

The problem is what values to put in the A and B vectors – to design the filter – that’s depends of your requirements. It can be designed in time or in frequency – I see Steve can help you with the filter responses.

Now about 8051 – this chip is not a good option for digital filtering, I have implemented filters with maximum of 3-4 dimension arrays for low frequency.

P.S. I was interrupted in the middle of my response – I agree with Peter – you must divide the problem.