Cutting glass is an exercise in preparation. 

First you need a flat table surface which has rug, resilient rubber, or other soft surface on it.

Second, the glass surface needs to be  clean.  Very clean.  Especially if you are attempting to cut old glass - make sure that the surface has no paint, grit, etc. on it.   Dirty glass and your cut will likely fail.

Third, measure your cut at least twice.  You know the adage: Measure twice, cut once.  Mark the glass with a wax or felt marker.  Get everything square.  And you will need a straight edge to run your glass cutter along.

Forth. lubricate the glass cutter (Fletcher is one brand I use) with a very light lubricant.  I use kerosene, Diesel  fuel, WD40, linseed oil, or even paint thinnner.

Fifth, apply quite a lot of hand (a lot of the pressure is in your thumb) pressure to the cutter - you will hear the surface of the glass rupturing/crunching - keep the rupturing noise even with even hand pressure as you run the cutter across the glass.  It is all in the noise.  Just from the sound I can tell if the cut is successful. The cut must be done in one continuous motion. You can't go back and re-do it. Start right at the very far edge of the glass, and in one continous motion run the cutter all the way to the close edge of the glass - and lighten up just a little as you reach the closest edge of the glass - but keep the glass rupturing noise - too much pressure right on the edge and you can split the glass and screw up your work.

This is like Yo Yo Ma with his bow....Or Tiger with his club.  very simple. we all can see what they are doing. right.

Sixth, turn the glass sheet over and use the ball on the end of the glass cutter to tap the glass just opposite the score mark.  Watch the glass and you can see a crack propogate through the glass from the side you are tapping to the side that was scored.   Do this tapping along the entire score mark. Make sure that the glass is cracking along the entire score.

Seventh, turn the glass sheet back over to the side you originally scored and bring the score line to the edge of the work table.   Now is the moment of truth:  using both hands evenly spread out on the glass edge - snap down firmly.  

You will snap the glass exactly along the score if all has been well prepared -

You can even do a serpentine cut like the one in the image above.  Get really good - circles are your thing. 

Good luck, Enjoy.  It's fun.

NB The cut in the image above was done by a friend of mine who had never cut glass before. I went through the above check list with my friend and did a few demo cuts as well.  When my friend took the cutter and swiped it over about 38 inches of the glass I could absolutely tell just from the noise of the rupturing glass that the cut was perfect.  My friend has a firm and concentrated grip. (see the white chalky edge of the top edge of the glass in the image - the chalky edge is from the tiny glass wheel rupturing the glass crystals in the surface of the glass).

Having an Attitute with the glass is important.  You are in charge!