First off welcome to HBT!
A yeast starter can certainly be made using sediment from a bottle or some yeast cake from a fermenter. I think what many people do is start out making starters with a vial of liquid yeast and sometimes that progresses into yeast washing and salvaging yeast from naturally carbonated bottles.
The idea of a yeast starter is to build the yeast colony up to proper pitching sizes for a given wort's specific gravity and volume. I believe the general rule of thumb is any 5 gallon batch 1.060 or below, a vial of yeast pitched straight into the wort is acceptable. Above that number, starters are recommended.
Think of a starter, kind of as a very small batch of beer but instead of beer the goal is to make more yeast. Make a wort mixture of around 1.035-1-040. You might think the higher OG the starter wort is the better, but it'll just stress out the yeast. Give them something gentle to chew on and they'll be much happier.
Here's a thread that that will help you with this.
You can search around HBT and find quite a few threads on this subject.
Happy Brewing and remember, if you are making beer you can make a starter!