Random error: These are fluctuations in a measurement (can be either positive or negative) due to the limitations of the thing you are using to take the measurement (and the person making the measurement). This basically means that an experimenter (yourself or any of us) do not take the same measurement in exactly the same way each time we make it, so there will be slight variations from measurement to measurement. Random errors can be reduced by taking a large number of measurements.
Systematic errors: These errors are exactly the same each time (they are in the same direction, ie either negative or positive) and are due to a "problem" in the whole experiment. Each time you take a reading it is off in the same direction by the same amount no matter how many times you make the measurement.
Some examples:
Random error: If you were to use an electronic balance to measure the mass of something and you did it three times you may get values of 45.56g, 45.58g, 45.54g. These are random errors and will decrease the more times you take the measurement.
Systematic error: Using the same example but this time your electronic balance has been tared incorrectly (for example it is tared 0.05g too high) so every measurement you make will be out by 0.05g, everytime, no matter how many measurements you make.
Note that these types of errors are associated with making measurements and are not the same as mistakes made by the experimentor. So forgetting to tare your balance for one of your measurements is not an error of measurement.
Maybe if you tell us some more information about your experiment we could be more specific.