The general problem about the classification of finite-dimensional Hopf
algebras (over $\mathbb{C}$) is widely open. I mention some general results.

Here

Zhu, Yongchang. Hopf algebras of prime dimension. Internat. Math. Res. Notices 1994, no. 1, 53--59. MR1255253 (94j:16072), link

it is proved that a Hopf algebra of prime dimension is isomorphic to a group
algebra. Here

Ng, Siu-Hung. Non-semisimple Hopf algebras of dimension $p^2$. J. Algebra 255 (2002), no. 1, 182--197. MR1935042 (2003h:16067), link

it is proved that the only Hopf algebras of dimension $p^2$ are the group
algebras and the Taft algebras. Hopf algebras of dimension $2p^2$ were also
classified by Hilgemann and Ng, see the following paper:

Hilgemann, Michael; Ng, Siu-Hung. Hopf algebras of dimension $2p^2$. J.
Lond. Math. Soc. (2) 80 (2009), no. 2, 295--310. MR2545253 (2010h:16080), link

Of course, these are not the only general results. For a good accound related
to the classification of Hopf algebras of a given dimension you may want to check
the following papers:

Beattie, Margaret. A survey of Hopf algebras of low dimension. Acta Appl. Math.
108 (2009), no. 1, 19--31. MR2540955 (2010i:16054), link

M. Beattie and G. A. García. Classifying Hopf algebras of a given dimension.
Preprint: arXiv:1206.6529

It is important to mention that it is very interesting to study the classification of certain families of finite-dimensional Hopf algebras. If this is your interest, maybe google can help.