You can read for instance the following article:

L. Chiantini - C. Ciliberto: "Weakly defective varieties", Transactions of the American Mathematical Society 354 (2001), p. 151 - 178,

which contains, among other results, the classification of weakly defective irreducible surfaces. This paper also gives a good list of references.

If you are looking for some techniques to prove weak non-defectivity, try reading

E. Ballico: "On the weak non-defectivity of Veronese embeddings of projective spaces", Central European Journal of Mathematics 3 (2005), p. 183-187.