To prevent your beer from becoming infected and tasting terrible, cleaning your carboy is very important.  When you have emptied a carboy of fermented beer there often is a ring around the top where the foam has touched.  This will require a brush such as this one.

Fill the carboy with a water and sanitizer solution following the directions for the sanitizer.  PBW is a good sanitizer and does not require a rinse after cleaning.  A couple scrubs from the brush and the residue should be gone.  Empty out the carboy in the sink or tub and place a sanitized stopper and airlock in the top.  This should keep it fairly clean for when you are ready to use it next.  You should always rinse the carboy with sanitizer before using it to make beer.  You should never use a cleaner such as bleach or soap as it can effect the taste and quality of the beer.

So you plunked down the big bucks (around $60) to upgrade from a hydrometer.  Refractometers have a couple advantages.  The first being ease of use.  The second is that they only use a couple of drops to get a gravity reading. 

The first thing you want to do is to calibrate your refractometer.  This can be done with the included solution or by using distilled water.  Lift the cover at the end of the refractometer and place a couple of drops on the blue colored prism.  Close the cover and look through the refractometer.  You might have to twist the eye piece to focus the image.  You should see what looks like a blue ruler.  Turn the adjustment screw until the white section stops and blue begins at the 0 level.

It should hopefully look like this when you have it calibrated. 

Now you are ready to take a reading.  Put a couple of drops of wort on the prism and close the top.  If the wort is hot, you should allow it to cool on the refractometer for several minutes.  This will ensure an accurate reading.  Where the blue meets the white is your %Brix.  Multiply this number by 4 and you’ve got your specific gravity.  Now you can use this to make sure you hit your target gravity or calculate the alcohol content of your beer.