This is short for bi-directional reflectance distribution function. A bdrf describes how light is reflected from a surface and in appleseed is connected to a material
The longer appleseed renders, the better the image quality will get. When an image has rendered long enough that extra render effort causes no perceivable difference, this is known as converging.
This is short for emission distribution function. Edf's describe how light is emitted from a surface. In appleseed, edf's are connected to a material.
A firefly is a common rendering artefact and can be caused by many things. The most common cause of fireflies in your render is having light sources that are hard for the renderer to find. Common reasons for a light source to be hard to find may be that it's very small, it's contribution is only visible after a number of light bounces or is only visible through a refractive object.
This one took a while for me to get my head around at first. In appleseed, when a ray hits an object it will loop through the lights in your scene and work out each light's contribution to that point on an object. It will then fire off another bounce ray to another point on a a different object and calculate how much light the second surface contributes light to the first as indirect light. In some renders, where there is little indirect light in the scene rather than just turning up the numbers of pixel samples, you can just then up the light samples. This means that more effort will be spent on calculating the direct contribution of lights than indirect.
Short for stochastic progressive photon mapping. Sppm is a rendering method where photons are emitted from lights. After hitting objects the renderer then connects these photons to the camera to render the final image. Sppm is currently the only practical way to render caustics or light sources with small or partially obscured emitters. It differs from regular photon mapping in that it can render scenes progressively with an interactive preview.