I've studied both chen style and yang style taiji and at the training sessions the first half is always spent doing various exercises, like
silk-reeling (in chen style) and neigong.
As I understand it, the term "silk-reeling" is best applied to Chen Style, but I know of many self-described Yang Stylists that practice "silk-reeling" exercises. According to the Chen style approach, all movement should proceed in spirals and these exercises help you focus on integrating the whole body to generate spiral movement in the limbs. I think Chen Style concentrates on originating the spirals from the dantian.
The term "neigong
" means something like "inner work/training/skill." It is used in opposition to "waigong" ("outer work/training/skill"). From what I understand, some use the term "neigong" for any type of training that focuses on "internal aspects" of training like qi cultivation or breathing.
From my perspective, there is not a clear separation between neigong and waigong, since both must contain aspects of the other; however, I would consider that exercises such as qi gong or standing or sitting mediation would be neigong. One of Yang Chengfu's Ten Essentials is "Unite inner and outer." This means that no matter what Tai Chi exercise you are doing, there ideally should be both neigong and waigong going on, even though you might focus more on one or the other.
As for describing the purpose of these exercises, it is hard to say without more specifics. The earlier posters gave a good range of some of them and some of their uses. I would say that most have multiple uses, depending on the level of your skill and understanding, just like doing the form. The form has so much going on, however, that it can be good to concentrate on specific things by doing simple exercises. It is also good to balance out your practice, so that doing a lot of rather external things, like fajin, is balanced with doing some rather internal things, like simple qi gong.
Specific goals of Tai Chi neigong might include helping to raise you spirit, calm and relax your mind, let the qi sink and concentrate in the dantian, and open up the meridians. Your eyes, breath, or voice can even get in the act, depending on what you are working on. Each of these things can be matched to something external. For instance, when we talk about issuing energy, we have both an internal and external perspective. To issue energy really well, you need to understand, train, and use both aspects together. To be able to use them together, it may be good to train them separately on occasion.