Asymptotics for the number of abelian groups of order at most $x.$

The problem was studied quite a bit; a complete summary will be complicated to give (in any case I cannot). A standard reference for classical results on this is A. Ivić "The Riemann Zeta-function: Theory and Applications" (1985); it seems there is a recent Dover edition. Ivić has various papers on this problem, too. The book has a chapter on this subject (14.5 Non-isomorphic abelian groups of a given order). At the moment I cannot recall what exactly is in there.

Some information related to this type of problem:

The correct asymptotics for $\sum_{n\in \mathbb{N}} a(n)$ were obtained by Erdős and Szekeres (1934), they proved: $$\sum_{n\le x} a(n) \sim A x + O(\sqrt{x})$$ with $A= \prod_{n \ge 2}\zeta(n)$.

By now there are more precise results known. For example the paper by Kendall and Rankin (1947), mentioned in another answer, gives $Ax + B x^{1/2} + O(x^{1/3} \log x )$, and developments continued, like the result mentioned by Roberto Pignatelli.

The to my knowledge latest improvement there is by Sargos and Wu (2000) $$Ax + Bx^{1/2} + Cx^{1/3} + O(x^{55/219} (\log x)^7 )$$ The first to get the third term of the main term was Richert (1952). For the historical development of the error see the introduction of a paper by Calderón (2003).

Related questions were also studied. For example, Kendall and Rankin (1947) showed that the "local densities" of $a(n)$ exist, that is $\sum_{n \le x, \, a(n)=k}1 \sim d_k x$ (good error terms are known). There are also estimates "in short intervals" so on $\sum_{ x \le n \le y, \, a(n)=k}1$.

For two recent papers on this see:

Emre Alkan, On the enumeration of finite abelian and solvable groups, J. Number Theory 101 (2003), no. 2, 404--423.

Ekkehard Krätzel, The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals, Arch. Math. (Basel) 91 (2008), no. 6, 518--525.

The introductions and references there will lead to various additional paper. (The first is freely accessible.)

References

C. Calderón: Asymptotic estimates on finite abelian groups. Publications De L’institut Mathematique, Nouvelle série, 74, 57-70 (2003).

P. Erdős, G. Szekeres: Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem (in German), Acta Litt. Sci. Szeged 7 (1934), 95--102; Zentralblatt 10,294. Free PDF

D. G. Kendall and R. A. Rankin, On the number of Abelian groups of a given order, Quart. J. Math., Oxford Ser. 18 (1947), 197--208.

Hans-Egon Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung. I, Math. Z. 56 (1952), 21--32.

P. Sargos and J. Wu, Multiple exponential sums with monomials and their applications in number theory, Acta Math. Hungar. 87 (2000), no. 4, 333--354.

One reference where the asymptotic result you are asking for was first established (I think), as well as some reasonable growth estimates for $a_n$, is

D.G.Kendall and R.A.Rankin, "On the number of Abelian groups of a given order", Quart. J. Math., Oxford Ser. 18 (1947), 197–208.

(full text: http://qjmath.oxfordjournals.org/content/os-18/1/197.full.pdf)

According to http://mathworld.wolfram.com/AbelianGroup.html it is a theorem of Srinivasan (1973), click on the link for details.