The A, B or C refer to weighting or filtering that is done at various frequencies. The "A" scale is an approximate representation of normal hearing. It is a standard used in medical, commercial and industrial applications. Someone concerned with hearing conservation might be doing measurements of specific areas or operations - this is normally done with the A weighting - and the readings are noted at dbA.
Equal sound energies at different frequencies will not all have the same apparent loudness because of the response of the human ear.
For example a 1KHz signal will sound much louder than a 50Hz siganl of the same power. The 'A' weighting curve tailors the response of the amplifier used in sound measurement so that the indicated sound levels give the apparent loudness of the sound - irrespective of frequency.
I'm not sure about 'C' weighting but seem to recall it compensates for how the ear responds to high intensity sound levels.