Depedning in the style of the tap, if you take off the head (i.e. the bit you turn) and sometimes a cover (often a chrome dome) where the spindle of the tap pops out (to which the head is attached) there is something called a stuffing box. This has (on older taps) hemp and paste in it. Usually it also has a nut (not the whole body of the tap, but just around this shaft). If you tighten this nut a very little (if you do it too much the tap will not turn when you re-assemble it) then this will more than likely cure the leak. On later taps this stuffing box does not exist, but there is an 'o' ring seal in there which may just be worn. Then you are probably looking at a complete replacement as they arenot usually easy to come by.
Pete K
January 2006