Good question! As a very rough rule of thumb, 20 seems to be a good score, in the sense that if your article gets an Altmetrics score of 20 or more, then it is getting more public attention than most other articles. However, the number is very sensitive to context. For instance, I could imagine that your research results could easily be presented in a way that would generate public interest and thus potentially get high Altmetric scores but would also have to compete with other high-scoring articles in the field. In contrast, I would be happily surprised if any of my pure maths articles were to get an Altmetric score of more than 0. It is therefore worth looking more closely at the Altmetric details of a particular article: the Altmetric data gives a detailed rundown of who was interested in your article and also how it is ranked in comparison with other articles in general, other articles from the same source, other articles of the same age, and so on. If your article gets more attention than 99% other articles from the same field, then that is worth bragging about, regardless of the particular Altmetric score.

In scholarly and scientific publishing, altmetrics are non-traditional metrics proposed as an alternative to more traditional citation impact metrics, such as impact factor and h-index, as on this site (RG) :)