Singeing fabric is melting the edges of material with heat. The point of singeing is to protect the fabric from unraveling and create a cleaner look and feel. I prefer to use a candle as my heat source because it is simple and easy.
Light a candle and cut your fabric. Bring the fabric just close enough to the flame to melt the edge but not catch fire. Rotate your fabric until all the edges are singed. That's it!
Some fabrics curl more than others. Just experiment a little with different fabrics to see what you like working with the most. I always look in the discount bins at fabric stores for great deals on small amounts of fabric for testing.
I've got perfect projects here and here for you to get started on with singeing.