At bare minimum, the starter culture must have a viable population of two bacteria, Lactobacillus Bulgaricus and Streptococcus thermophilus. These two bacteria work together to consume the lactose and produce the acid that sets the yogurt, as well as the compounds that give yogurt its myriad of tangy, sour, and tart flavors. The lactobacilli bacteria are responsible for most of the acidity, while the streptococci and other strains make some of the more interesting, delicate flavors. A starter heavy in lactobacilli strains will produce an acidic yogurt that will be sure to set, but it may lack the grassy and cheese-like flavors you can get from more diverse cultures.
You can get your starter population from a variety of places, the easiest being a container of store-bought yogurt – just make sure to choose a yogurt that you like with no added flavors or sweeteners. You can also use a specially formulated yogurt starter, which usually comes in the form of a freeze dried (or lyophilized) powder. Most of the ones that you are likely to find at your grocery store are going to be heavy in lactobacillus and will give you a very tart yogurt. There are also a number of websites selling traditional or heirloom cultures that contain a more diverse population of bacteria.
The amount of starter that you use can have a big effect on the result, especially if your starter is from your previous batch. Some of the bacteria involved in the fermentation become inactive after a certain level of acidity is reached, so if you use too much starter then the mixture might be too acidic to give those bacteria a chance to make their contribution. In The Art of Fermentation, Sandor Ellix Katz explains that sometime he gets a thicker yogurt when using less starter, probably for the reason I described above. He recommends about a tablespoon per quart, or around 5% by volume, but notes that some references say you can go as low as 2%.